Semi-Infinite Homology of Floer Spaces Piotr Suwara

Semi-infinite Homology of Floer Spaces by Piotr Suwara Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2020 © Piotr Suwara, MMXX. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Author...... Department of Mathematics August 6, 2020

Certified by...... Tomasz S. Mrowka Professor of Mathematics Thesis Supervisor

Accepted by ...... Davesh Maulik Chairman, Department Committee on Graduate Theses 2 Semi-infinite Homology of Floer Spaces by Piotr Suwara

Submitted to the Department of Mathematics on August 6, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics

Abstract This dissertation presents a framework for defining Floer homology of infinite-dimensional spaces with a functional. This approach is meant to generalize the tradi- tional constructions of Floer homologies which mimic the construction of the Morse- Smale-Witten complex. To define Floer homology we use cycles modelled on infinite- dimensional manifolds with corners, as described by Maksim Lipyanskiy, where the key is to impose appropriate compactness and polarization axioms on the cycles. We separate and carefully inspect these two types of axioms, paying special attention to correspondences, generalizing the definition of a polarization as well as axiomatizing the notion of a family of perturbations. The latter is used to define an intersection pairing and maps induced on Floer homology by correspondences. Moreover, we prove suspension isomorphisms and prove that this Floer homology agrees with Morse homology for finite-dimensional manifolds with a Palais-Smale functional. Finally, we explain how to apply this framework to Seiberg-Witten-Floer theory, defining Floer homology groups associated to rational homology spheres and their spinc-structures. Importantly, we prove moduli spaces of solutions to Seiberg-Witten equations induce maps on Floer homology in a functorial fashion.

Thesis Supervisor: Tomasz S. Mrowka Title: Professor of Mathematics

3 4 Acknowledgments

Firstly, I would like to express my deep gratitude to my advisor Tomasz Mrowka for his support and advice during my graduate study at MIT. He has shared with me a tremendous amount of knowledge and understanding of gauge theory and given important insights about his previous work with Peter Ozsváth which is the founda- tion of this thesis, along with later work carried out by Maksim Lipyanskiy. Most importantly, Tom has deepened my understanding and appreciation of mathematics immensely, and has been very supportive of my math efforts. I am indebted to Paul Seidel and Matthew Stoffregen for serving on my thesis committee and to Maksim Lipyanskiy for sharing his ideas about the development of the theory of semi-infinite cycles. I am also particularly grateful to Yakov Barchenko- Kogan, Jesse Freeman, Sherry Gong, MacKee Krumpak, Zhenkun Li, Francesco Lin, Jianfeng Lin, Langte Ma, Haynes Miller, Kevin Sackel, Donghao Wang and Boyu Zhang for many insightful conversations about math and Ph.D. studies. Many people helped me to become a mathematician before I came to MIT. While the full list would be too long to include, I want to give special thanks to Jerzy Konarski and Maciej Borodzik for going out of their way to help me grow mathemat- ically. Finally, my journey would have not been possible without the help of my parents, whom I want to thank for their unconditional love and support.

5 6 Contents

1 Introduction 9 1.1 Floer Spaces in Topology ...... 9

2 Floer Spaces 13 2.1 Semicompact maps ...... 14 2.2 Polarized Hilbert spaces ...... 17 2.3 Linear correspondences ...... 30 2.4 Polarized manifolds ...... 36 2.5 Boundary of a manifold with corners ...... 45 2.6 Floer spaces and correspondences ...... 47

3 Homology of Floer Spaces 53 3.1 Defining homology ...... 53 3.2 Homology in finite dimensions ...... 55 3.3 Perturbing chains ...... 60 3.4 Suspension isomorphisms ...... 67 3.5 Perturbing functionals ...... 72

4 Seiberg-Witten Floer Spaces 75 4.1 Seiberg-Witten-Floer spaces and gauge actions ...... 75 4.2 Moduli spaces are cycles ...... 79 4.3 Moduli spaces are correspondences ...... 84 4.4 Existence of admissible perturbations ...... 98

7 4.5 Invariance under perturbations of the functional ...... 99

A Appendix 101 A.1 Sobolev spaces ...... 101 A.2 Infinite-dimensional manifolds ...... 104 A.3 Weakly convergent operators ...... 106 A.4 Regularity of solutions ...... 106 A.5 Unique continuation ...... 109

8 Chapter 1

Introduction

1.1 Floer Spaces in Topology

Use of gauge theory in topology was pioneered by Donaldson in 1983 when he proved that a definite intersection form of a simply-connected oriented compact smooth manifold is diagonalizable over the integers [Don83]. To prove it, Donaldson analyzed the moduli space of anti-self-dual connections on a principal SU(2)-bundle over the four-manifold. Later in the 1980’s, Andreas Floer used Morse theory, most notably the construction of the Morse-Smale-Witten complex (cf. [Wit82, Flo87, Flo89b]), in an infinite-dimensional setting to develop the Lagrangian Floer homology which he used to prove Arnold’s conjecture for a class of symplectic manifolds [Flo88b,Flo88c, Flo89a]. Soon after that he used the same ideas to construct Instanton Floer homology for 3-manifolds by studying the Morse-Witten complex associated to the Yang-Mills functional [Flo88a]. In turn, Donaldson realized [DK90] that cobordism induce maps between the respective complexes providing the theory with a TQFT-like structure. Since then, various flavors and generalizations of Instanton Floer homology have been defined and successfully applied to the study of low-dimensional topology (e.g., [AB96, Frø02, KM10, KM11b, KM11a, Mil19]). Kronheimer and Mrowka used these ideas to define yet another flavor of Floer homology, called Monopole homology [KM07] (or Seiberg-Witten-Floer homology), utilizing the Chern-Simons-Dirac functional and the associated Seiberg-Witten equations. Aiming to provide a more com-

9 putable version of Monopole homology, Ozsvath and Szabo created Heegaard Floer homology [OS04a, OS04b, OS04c] which is a variant of Lagrangian Floer homology and has been proven to be isomorphic to Monopole homology.

All of these theories rely on emulating the construction of the Morse-Smale-Witten complex for an infinite-dimensional space X with a function ℒ : X → R, (usually 2 called a functional). The space X is typically modelled as a space of 퐿푘 maps from 2 a manifold 푌 , for instance a space of 퐿푘 sections of a bundle over 푌 and the downward 퐿2 gradient flow of ℒ determines a system of PDEs on 푌 ×R. The translation-invariant solutions to this system are critical points of ℒ and provide generators of the Morse- Smale-Witten complex, while non-constant solutions approaching critical points at their ends are used to construct the differential. The associated homology is shown to be independent of various choices (e.g., of a metric on 푌 ) and is generally called “Floer homology”.

In the case of Instanton and Monopole theories, the 4-dimensional equations on

푌 × R can be defined on any 4-manifold. For a cobordism 푊 with 휕푊 = −푌0 ⊔ 푌1 * one can construct s 푊 = 푊 ∪ 푌0 × (−∞, 0] ∪ 푌1 × [0, ∞) and by considering the moduli spaces of solutions on 푊 * one defines a chain map between the chain complexes associated to 푌0 and 푌1 and thus a map between their Floer homologies. In particular, any 4-manifold 푋 with boundary 휕푋 = 푌 gives rise to an element in the Floer homology of 푌 .

In this thesis I present and develop a construction of Floer homology using “semi- infinite chains”, which are to be understood as maps from a variant of infinite dimensional manifolds with corners to the Floer space, satisfying some compatibility assumptions with respect to ℒ, which we call a Floer functional (while X with such a functional will be called a Floer space) Already in 1988, Atiyah suggested the possibility of using such chains to define Floer homology [Ati88]. The idea for this construction stems from an unpublished work of Mrowka and Ozsvath and has already been presented in the Ph.D. thesis of Max Lipyanskiy [Lip08]. Such a construction is motivated by analytical problems arising in the aforementioned Floer theories, which we will call Morse-Floer theories. Indeed, even proving that the differential in the

10 Morse-Witten chain complex squares to zero requires a subtle procedure of compactifying the moduli spaces of trajectories. A much more serious problem is the definition of equivariant homology group in case there is a Lie group 퐺 acting on the Floer space X with ℒ equivariant. Such action can come from the symmetries innate to the theory (e.g., 푆1-action on the Seiberg-Witten-Floer space) or can be induced by an action on the underlying 3-manifold 푌 . In Morse theory one needs to perturb the functional ℒ to obtain transversality needed for the construction to work. However, it is possible that there is no equivariant perturbation which would yield the expected transversality conditions. Multiple authors are nowadays trying to define equivariant Floer homologies (cf. [AB96, SS10, Hen12, Mil19]). We expect such a construction to be simplified with the use of semi-infinite cycles.

Therefore, the semi-infinite cycle construction is expected to yield definitions of new Floer theories, in particular equivariant Floer theories. We hope that analo- gous constructions will let us define other invariants of Floer spaces coming from algebraic topology, like K-theory, without having to construct a homotopy type first. Finally, it might become a convenient framework for increasing our understanding of Floer theories, in particular for proving relationships between various flavors of Floer homologies.

In Chapter 2 I lay down foundations for the development of homology theory of Floer spaces. The discussion consists of two components: analyzing the appropriate compactness properties (semicompactness) that are required of semiinfinite chains and analyzing the behavior of their differentials. Polarizations are introduced to both make sure that intersections of chains and chains in the dual Floer space are finite-dimensional and to introduce gradings on chains. Correspondences between Floer spaces are defined which are supposed to induce maps on homology. ThenI proceed to presenting the definition of Floer homology using semi-infinite cycles and maps induced by correspondences in Chapter 3. There I also present computations in finite dimensions and prove suspension isomorphisms. I discuss how to perturb chains to achieve transversality while making sure the perturbed map remains a chain homotopic to the original one. Moreover, I establish a class of perturbations of the

11 Floer functional which do not change homology. Finally, in Chapter 4 I define the Seiberg-Witten Floer spaces for rational homology spheres and explain how cobordisms between them give correspondences between the respective Floer spaces. I prove functoriality of the maps induced by correspondences and invariance of semiinfinite homology under perturbations. Most of the results are based on Max Lipyanskiy’s work [Lip08], but explained in much more detail, and some technical errors are cor- rected.

12 Chapter 2

Floer Spaces

We start by laying out the theory of Floer spaces with the goal of defining semi-infinite chains. The definition will consist of two aspects: firstly, we need to find appropriate compactness conditions for chains modelled on infinite-dimensional Hilbert spaces (as opposed to finite-dimensional), allowing them to “extend to −∞” (as the functional may not be bounded). Such maps will be called semicompact. The latter condition may be motivated by the example of R2 with the functional ℒ(푥, 푦) = −푥2 + 푦2, where we want the homology in degree 1 to be generated by the 푥-axis, which is the unstable manifold of the critical point at (0, 0).

Secondly, to make sure transverse intersections of chains for ℒ and for −ℒ are finite-dimensional, we require chains to be polarized. Given a decomposition 푇 푀 = 푇 +푀⊕푇 −푀 for a Hilbert manifold 푀, polarization means that the chains 휎 : 푃 → 푀 need to be “aligned to 푇 −푀”, meaning the components 퐷±휎 : 푇 푃 → 푇 ±푀 of the differential 퐷휎 have to be compact ( 퐷+휎 ) and Fredholm ( 퐷−휎), respectively. We generalize the notion of polarization to make sure finite-dimensional discontinuities in the decomposition can occur.

We also need to study correspondences 퐹 : 푍 → 푀 × 푁 with respect to both of these aspects. We define correspondences in such a way that if 휎 : 푃 → 푀 is a chain, then the fiber product 휎 ×푀 퐹 : 푃 ×푀 푍 → 푁 is a chain as well.

13 2.1 Semicompact maps

In this section we define semicompact maps and correspondences, generalizing the notion of a compact chain to the Floer-theoretic setting.

Definition 2.1 (Floer functional). Let 푋 be a metric space equipped with its usual (strong) topology and a choice of weak topology (which is required to be coarser than the strong topology, but still Hausdorff).

We call a continuous function ℒ : 푋 → R a Floer functional on 푋 if it satisfies the following axiom:

Axiom F1. ℒ is bounded on weakly precompact sets.

The functional −ℒ will be called the dual Floer functional to ℒ and denoted also as ℒ∨.

Remark 2.2. When we say that a function is continuous, we mean it is continuous with respect to the strong topology; a function continuous with respect to the weak topology will be called weakly continuous. Similarly, we will use precompact as well as weakly precompact sets, convergent as well as weakly convergent sequences, etc.

Example 2.3. Any continuous function 푓 : 푀 → R on a compact finite-dimensional manifold 푀 is a Floer functional.

+ − Example 2.4. Let 퐻 = 퐻 ⊕퐻 be a direct sum of Hilbert spaces. Then ℒ(푣+ + 푣−) = 2 2 |푣+| − |푣−| is a Floer functional.

Definition 2.5 (semi-compact map). Let ℒ be a Floer functional on 푋 and 푃 a topological space. A map 휎 : 푃 → 푋 is called semi-compact with respect to ℒ if it satisfies the following:

Axiom M1. On the weak closure of 휎(푃 ), the function ℒ is bounded above and upper semi-continuous for the weak topology.

14 Axiom M2. Any subset 푆 ⊂ im 휎 on which ℒ is bounded is precompact for the weak topology.

Axiom M3. If 휎(푥푖) ⇀ 푦 and lim ℒ(휎(푥푖)) = ℒ(푦), then 푥푖 has a (strongly) convergent subsequence.

Definition 2.6 (semi-compact correspondence). Let ℒ푖 : 푋푖 → R be Floer 푠 푡 ∨ functionals, 푖 = 1, 2. A map 푓 = (푓 , 푓 ): 푍 → 푋1 × 푋2 is called a semi-compact correspondence if it satisfies the following:

Axiom C1. On the weak closure of im 푓, the function ℒ = ℒ2 −ℒ1 is bounded above and upper semi-continuous for the weak topology.

푠 푡 Axiom C2. If 푓 (푧푖) is weakly precompact and ℒ2(푓 (푧푖)) is bounded below, then 푡 푓 (푧푖) is weakly precompact.

푠 푡 Axiom C3. If 푓 (푧푖) → 푥 and 푓 (푧푖) ⇀ 푦 and ℒ(푓(푧푖)) → ℒ(푥, 푦), then 푧푖 has a (strongly) convergent subsequence.

∨ Remark 2.7. Notice that a semi-compact map 푓 : 푍 → 푋1 × 푋2 (with respect to

−ℒ1 + ℒ2) is also a correspondence from 푋1 to 푋2.

Proposition 2.8 (semi-compact correspondence composition). Let 푓 : 푍 →

∨ ∨ 푋0 × 푋1 and 푔 : 푊 → 푋1 × 푋2 be semi-compact correspondences with respect to

Floer functionals ℒ푖 : 푋푖 → R. ∨ Then the composition 푔 ∘ 푓 := 푓 ×푋1 푔 : 푍 ×푋1 푊 → 푋0 × 푋2, where 푍 ×푋1 푊 = {(푧, 푤) ∈ 푍 × 푊 : 푓 푡(푧) = 푔푠(푤)} is a semi-compact correspondence.

푠 푡 푡 푠 Proof. Axiom C1. Let (푓 (푧푖), 푔 (푤푖)) ⇀ (푎, 푐) with 푓 (푧푖) = 푔 (푤푖).

15 푡 푡 By Axiom F1, ℒ2(푔 (푤푖)) is bounded. Axiom C1 for 푔 implies ℒ2(푔 (푤푖)) − 푠 푠 푡 ℒ1(푔 (푤푖)) is bounded above and therefore ℒ1(푔 (푤푖)) = ℒ1(푓 (푧푖)) is bounded below. 푡 푠 Axiom C2 for 푓 implies now that {푓 (푧푖) = 푔 (푤푖)} is weakly precompact. After taking a subsequence, we get 푓(푧푖) ⇀ (푎, 푏) and therefore also 푔(푤푖) ⇀ (푏, 푐).

Since ℒ2(푐) − ℒ0(푎) = ℒ2(푐) − ℒ1(푏) + ℒ1(푏) − ℒ0(푎), Axiom C1 for 푓, 푔 implies that ℒ2 −ℒ0 is bounded above and weakly upper semi-continuous on the weak closure of im(푔 ∘ 푓).

푠 푡 푡 푠 푡 Axiom C3. Assume 푓 (푧푖) → 푎, 푔 (푤푖) ⇀ 푐, 푓 (푧푖) = 푔 (푤푖) and ℒ2(푔 (푤푖)) − 푠 푠 ℒ0(푓 (푧푖)) → ℒ2(푐) − ℒ0(푎). First, by the continuity of ℒ0, we get that ℒ0(푓 (푧푖)) → 푡 푡 ℒ0(푎) and thus ℒ2(푔 (푤푖)) → ℒ2(푐). Therefore {ℒ2(푔 (푤푖))} is bounded, and it follows 푠 푡 푡 that {ℒ1(푔 (푤푖)) = ℒ1(푓 (푧푖))} is bounded below. Thus, by Axiom C2, {푓 (푧푖) = 푠 푔 (푤푖)} is weakly precompact; without loss of generality, we can assume it is weakly convergent to 푏. Finally, by Axiom C1, we get inequalities

푡 푠 lim sup(ℒ1(푓 (푧푖)) − ℒ0(푓 (푧푖))) ≤ ℒ1(푏) − ℒ0(푎) 푖→∞ 푡 푠 lim sup(ℒ2(푔 (푤푖)) − ℒ1(푔 (푧푖))) ≤ ℒ2(푐) − ℒ1(푏) 푖→∞

On the other hand, we assumed that

푡 푠 푡 푠 lim (ℒ1(푓 (푧푖)) − ℒ0(푓 (푧푖)) + ℒ2(푔 (푤푖)) − ℒ1(푔 (푤푖))) 푖→∞ 푡 푠 = lim (ℒ2(푔 (푤푖)) − ℒ0(푓 (푧푖))) 푖→∞

= ℒ2(푐) − ℒ0(푎) and this implies that actually

푡 푠 lim (ℒ1(푓 (푧푖)) − ℒ0(푓 (푧푖))) = ℒ1(푏) − ℒ0(푎) 푖→∞ 푡 푠 lim (ℒ2(푔 (푤푖)) − ℒ1(푔 (푧푖))) = ℒ2(푐) − ℒ1(푏). 푖→∞

푡 푠 Using Axiom C3 for 푓 and 푔 consecutively, we find first that 푓 (푧푖) = 푔 (푤푖) has 푡 a strongly convergent subsequence and then that 푔 (푤푖) has a strongly convergent

16 subsequence.

푠 푡 Axiom C2. Assume {푓 (푧푖)} is weakly precompact, {ℒ2(푔 (푤푖))} is bounded 푡 푠 푠 푡 below and 푓 (푧푖) = 푔 (푤푖). By Axiom C1 for 푔, {ℒ1(푔 (푤푖)) = ℒ1(푓 (푧푖))} is bounded 푡 푠 below and thus, by Axiom C2, {푓 (푧푖) = 푔 (푤푖)} is weakly precompact. The same 푡 Axiom for 푔 implies that {푔 (푤푖)} is weakly precompact.

∨ Corollary 2.9. If 휎 : 푃 → 푋0 is a semi-compact map and 푓 : 푍 → 푋0 × 푋1 is

a semi-compact correspondence, then 푓 ∘ 휎 : 푍 ×푋0 푃 → 푋1 is a semi-compact map.

2.2 Polarized Hilbert spaces

In this section, we define a category of polarized Hilbert spaces. We also analyze negative maps, which later will serve as the model for differential maps of Floer cycles. Our discussion is based on [CJS95]. We first fix some notations. Throughout the article by a “Hilbert space” wemean a real, separable Hilbert space, unless noted otherwise. For 퐻 a Hilbert space (resp.

퐻1, 퐻2 Hilbert spaces) we denote by 퐵(퐻) or by Hom(퐻, 퐻) (resp. by Hom(퐻1, 퐻2)) the set of all continuous linear maps (operators) 퐻 → 퐻 (resp. 퐻1 → 퐻2). We will use the following notations for some of its subsets:

• ℱ(퐻) (resp. ℱ(퐻1, 퐻2)) denotes the set of Fredholm operators,

• 풦(퐻) (resp. 풦(퐻1, 퐻2)) denotes the set of compact operators,

• GL(퐻) (resp. GL(퐻1, 퐻2)) denotes the set of invertible operators.

Whenever it does not lead to confusion, we will use abbreviated notations ℱ, 풦, GL.

Definition 2.10 (polarization). A polarization of a Hilbert space 퐻 is a set 풥 ⊂ 퐵(퐻) of continuous operators 퐽 : 퐻 → 퐻 congruent modulo compact operators such that 퐽 2 ∈ 1 + 풦(퐻), i.e., 퐽 2 = 1 modulo compact operators. We call the pair (퐻, 풥 ) a polarized space.

17 If 퐽 ∈ 퐵(퐻) satisfies 퐽 2 = 1 modulo compact operators, we denote by [퐽] the polarization induced by 퐽, i.e. [퐽] = 퐽 + 풦(퐻). By 풥 ∨ we denote the dual polarization, 풥 ∨ = −풥 .

When it does not cause confusion, we will simply call 퐻 a polarized space.

Definition 2.11. For any Hilbert space 퐻, we will call 풥+ = [+1] (resp. 풥− = [−1]) the positive (resp. negative) polarization of 퐻. We will call the polarization proper if it is neither positive or negative.

2 + − A choice of 퐽 ∈ 풥 such that 퐽 = 1 induces a decomposition 퐻 = 퐻퐽 ⊕ 퐻퐽 into ± the ±1-eigenspaces of 퐽, 퐻퐽 = ker(1 ∓ 퐽). One also obtains complementary (but not ± 1±퐽 ± ± ∓ necessarily orthogonal) projections Π퐽 = 2 : 퐻 → 퐻퐽 with ker Π퐽 = 퐻퐽 . On the other hand, given a decompostion 퐻 = 퐻+ ⊕ 퐻− into closed subspaces one can define 퐽|퐻+ = 1 and 퐽|퐻− = −1 to obtain a polarization 풥 .

Definition 2.12. For 퐻 = 퐻+ ⊕퐻−, 퐽 and 풥 as above, we say that 퐻 = 퐻+ ⊕퐻− represents 풥 and is induced by 퐽.

Lemma 2.13. Each polarization 풥 has a representative 퐽 ∈ 풥 such that 퐽 2 = 1 exactly (not just modulo compact operators).

In other words, every polarization 풥 of 퐻 can be represented by a decomposition 퐻 = 퐻+ ⊕ 퐻− by taking 퐻± to be the ±1-eigenvalue of 퐽 for which 퐽 2 = 1. Note also that in this case the dual polarization −풥 is represented by the decomposition 퐻 = 퐻− ⊕ 퐻+.

Proof. Since 퐽 2 = 1 + 퐾 for some compact operator 퐾, therefore the spectrum 휎(퐽 2)

2 of 퐽 is equal to 1 + 휎(퐾), while 휎(퐾) is a countable sequence of eigenvalues 휆푖 → 0.

18 In other words, 휎(퐽 2) is at most countable and the only possible accumulation point is 1 ∈ C. Since 휎(퐽)2 ⊂ 휎(퐽 2), we conclude that 휎(퐽) is also at most countable and the only possible accumulation points are +1, −1 ∈ C. Moreover, the generalized eigenspace of 퐽 for any 휆 ∈ 휎(퐽) ∖ {±1} is finite dimensional because the generalized eigenspace of 퐾 for any (휆2 −1) ∈ 휎(퐾) is finite-dimensional. We can thus decompose 휎(퐽) = 휎+(퐽)⊔휎−(퐽) with 휎±(퐽) having only one accumulation point ±1 (if any). By

− taking a counterclockwise loop 훾 enclosing a disk 퐷훾 with 퐷훾 ∩ 휎(퐽) = 휎 (퐽) we can − 1 ∮︀ 1 − − construct the Riesz projection Π = 2휋푖 훾 푧−퐽 푑푧. We get that 휎(퐽Π |im Π− ) = 휎 (퐽) and 퐽Π− = Π−퐽 and therefore (퐽+1)Π− has spectrum with at most one accumulation point, at 0, thus is compact. Similarly for Π+ = 1 − Π− we get that (퐽 − 1)Π+ is compact. Finally, take 퐽˜ = Π+ − Π−. We get 퐽 − 퐽˜ = 퐽(Π+ + Π−) − (Π+ − Π−) = (퐽 − 1)Π+ + (퐽 + 1)Π− which is compact. So [퐽] = [퐽˜] and 퐽˜2 = 1, which was to be

proven.

If 퐻 is polarized (resp. 퐻1, 퐻2 are polarized) then we can identify a class of operators respecting this additional structure.

Definition 2.14. Let 퐻 be a polarized Hilbert space with polarization 풥 (resp.

퐻1, 퐻2 be polarized with polarizations 풥1, 풥2). We say that 퐴 ∈ ℱ(퐻) (resp. 퐴 ∈

ℱ(퐻1, 퐻2)) is polarized if for any 퐽 ∈ 풥 (resp. for any 퐽1 ∈ 풥1, 퐽2 ∈ 풥2) the

commutator [퐴, 퐽] = 퐴퐽 − 퐽퐴 (resp. 퐴퐽1 − 퐽2퐴) is compact.

We denote the set of all such 퐴 ∈ ℱ(퐻) (resp. 퐴 ∈ ℱ(퐻1, 퐻2), 퐴 ∈ GL(퐻),

퐴 ∈ GL(퐻1, 퐻2)) by ℱ푟푒푠(퐻) (resp. 퐴 ∈ ℱ푟푒푠(퐻1, 퐻2), 퐴 ∈ GL푟푒푠(퐻), 퐴 ∈

GL푟푒푠(퐻1, 퐻2)).

Note that the “for all” quantifier in the definition above can be replaced with “for

any”, i.e., it is necessary and sufficient to check the compactness of 퐴퐽1 − 퐽2퐴 for one

choice of the pair 퐽푖 ∈ 풥푖.

19 Remark 2.15. Our definitions of restricted operators differ from the definitions of Presley and Segal [PS86] since we are using the class of compact operators instead of the more restricted class of Hilbert-Schmidt operators. However, the results about homotopy of the classes of operators and Grassmanians remain the same, along with the proofs, the key fact being that the space of compact operators 풦(퐻1, 퐻2) is contractible.

+ − Proposition 2.16. Let 퐻푖 = 퐻푖 ⊕ 퐻푖 be decompositions induced by 퐽푖 ∈ 풥푖, for ⎛ ⎞ 푎 푏 ⎜ ⎟ 푖 = 1, 2. Then any 퐴 ∈ Hom(퐻1, 퐻2) can be written in matrix form 퐴 = ⎝ ⎠ : 푐 푑 + − + − 퐻1 ⊕ 퐻1 → 퐻2 ⊕ 퐻2 and 퐴 ∈ ℱ푟푒푠(퐻1, 퐻2) if and only if 푏, 푐 are compact and 푎, 푑 are Fredholm.

Moreover, if 퐴 ∈ ℱ푟푒푠 is Fredholm then ind 퐴 = ind 푎 + ind 푑 and any Fredholm inverse 퐵 ∈ ℱ(퐻2, 퐻1) is polarized, i.e., 퐵 ∈ ℱ푟푒푠(퐻2, 퐻1). ⎛ ⎞ 0 푏 ⎜ ⎟ Proof. We have 퐴퐽1 − 퐽2퐴 = −2 ⎝ ⎠ which is compact if and only if 푏, 푐 are 푐 0 compact. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 푎 푏 0 푏 0 푏 푎 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ If 퐴 = ⎝ ⎠ is Fredholm and ⎝ ⎠ is compact, then 퐴 − ⎝ ⎠ = ⎝ ⎠ 푐 푑 푐 0 푐 0 0 푑 is Fredholm, which implies that 푎, 푑 are Fredholm (c.f. Proposition 2.43) and that ⎛ ⎞ 푎 0 ⎜ ⎟ ind(퐴) = ind ⎝ ⎠ = ind(푎) + ind(푑). On the other hand, if 푎, 푑 are Fredholm, the 0 푑 same reasoning shows that 퐴 is Fredholm. ⎛ ⎞ 푒 0 ⎜ ⎟ Let 푒, 푓 be Fredholm inverses of 푎 and 푑. Then ⎝ ⎠ is a Fredholm inverse to 퐴, 0 푓 belongs to ℱ푟푒푠(퐻2, 퐻1) and any other Fredholm inverse differs from it by a compact operator.

Proposition 2.17. Let 풥1, 풥2 be two polarizations of 퐻. The following are equivalent:

1. 풥1 = 풥2.

2. For all 퐽1 ∈ 풥1, 퐽2 ∈ 풥2 the commutator [퐽1, 퐽2] is compact (i.e., 퐽1 ∈ GL푟푒푠(퐻, 풥2)).

20 3. For some 퐽1 ∈ 풥1, 풥2 ∈ 풥2 the commutator [퐽1, 퐽2] is compact.

+ − 4. For any choice of decompositions 퐻푖 = 퐻푖 ⊕퐻푖 representing 풥푖, the projections − + − + induced from polarizations Π2 | + = (1 − 퐽2)/2 : 퐻1 → 퐻2 and Π2 | − = 퐻1 퐻1 − + + + + − (1 + 퐽2)/2 : 퐻1 → 퐻2 are compact, while Π2 | + : 퐻1 → 퐻2 and Π2 | − : 퐻1 퐻1 − − 퐻1 → 퐻2 are Fredholm.

+ − 5. For some choice of decompositions 퐻푖 = 퐻푖 ⊕ 퐻푖 representing 풥푖, the pro- − + − jections induced from polarizations Π2 | + = (1 − 퐽2)/2 : 퐻1 → 퐻2 and 퐻1 + − + + + + Π2 | − = (1 + 퐽2)/2 : 퐻1 → 퐻2 are compact, while Π2 | + : 퐻1 → 퐻2 퐻1 퐻1 − − − and Π | − : 퐻 → 퐻 are Fredholm. 2 퐻1 1 2

Proof. Condition (1) implies (2) since, modulo compact operators, [퐽1, 퐽2] = 퐽1퐽2 − 2 2 퐽2퐽1 ≡ 퐽1 + 퐽1퐽2 − 퐽2퐽1 − 퐽2 = (퐽1 − 퐽2)(퐽1 + 퐽2) ≡ 0(퐽1 + 퐽2) = 0.

Condition (2) and (3) are equivalent because operators in 풥푖 differ by compact operators. Condition (2) implies (4) by Proposition 2.16. Condition (4) implies (5) directly.

− Condition (5) implies (1) because (퐽1 − 퐽2)| + = (1 − 퐽2)| + = 2Π2 and (퐽1 − 퐻1 퐻1 + 퐽2)| − = −(1 + 퐽2)| − = −2Π are compact by (4). 퐻1 퐻1 2

There is a canonical polarization on the direct sum of two polarized spaces:

Definition 2.18 (direct sum of polarizations). If (퐻1, 풥1) and (퐻2, 풥2) are polarized Hilbert spaces and 퐻 = 퐻1 ⊕ 퐻2 we denote by 풥1 ⊕ 풥2 the polarization

퐽1 + 퐽2 + 풦(퐻) for any 퐽1 ∈ 풥1, 퐽2 ∈ 풥2.

If 퐻 = 퐻1 ⊕ 퐻2 with 퐻, 퐻1, 퐻2 polarized then the polarization on 퐻 comes from the polarizations on 퐻1 and 퐻2 as a direct sum if and only if 퐽 − 퐽1 ⊕ 퐽2 ∈ 풦(퐻) or equivalently [퐽, 퐽1 ⊕퐽2] ∈ 풦(퐻) for some representatives 퐽, 퐽1, 퐽2 (Proposition 2.17).

We may also ask whether having polarizations on 퐻 and 퐻1 induces a polarization on 퐻2. In such case we must have 퐽 −퐽1 ⊕퐽2 ∈ 풦(퐻) and thus 퐽Π퐻1 −퐽1 ⊕0 ∈ 풦(퐻)

21 and Π퐻1 퐽 − 퐽1 ⊕ 0 ∈ 풦(퐻). It also follows that [퐽, 퐽1 ⊕ 0] ≡ [퐽1 ⊕ 퐽2, 퐽1 ⊕ 0] = 0 modulo 풦(퐻). These two necessary conditions are actually equivalent and sufficient for existence of such 풥2:

Proposition 2.19. Let 풥 and 풥1 be polarizations on 퐻 = 퐻1 ⊕ 퐻2 and 퐻1, respectively, represented by 퐽 and 퐽1. Then the following conditions are equivalent:

1. 퐽Π퐻1 − 퐽1 ⊕ 0 ∈ 풦(퐻) and Π퐻1 퐽 − 퐽1 ⊕ 0 ∈ 풦(퐻),

2. [퐽, 퐽1 ⊕ 0] ∈ 풦(퐻),

3. there exists a polarization 풥2 such that 풥 = 풥1 ⊕ 풥2.

Proof. We already explained how (3) implies (1) and (2).

Assume (1) and (2) hold. Take 퐽2 = Π퐻2 퐽|퐻2 . We claim 풥2 = [퐽2] is the desired polarization. Firstly, 퐽Π퐻2 = 퐽 − 퐽Π퐻1 ≡ 퐽 − 퐽1 ⊕ 0 modulo 풦(퐻). Moreover, 2 퐽(퐽1 ⊕ 0) = 퐽Π퐻1 (퐽1 ⊕ 0) ≡ (퐽1 ⊕ 0) ≡ id퐻1 modulo 풦(퐻). We get

2 2 2 (퐽Π퐻2 ) = 퐽 + 퐽1 ⊕ 0 − 퐽(퐽1 ⊕ 0) − (퐽1 ⊕ 0)퐽

≡ id퐻 + id퐻1 ⊕ 0 − 2퐽(퐽1 ⊕ 0)

≡ id퐻 + id퐻1 ⊕ 0 − 2(퐽1 ⊕ 0)(퐽1 ⊕ 0)

≡ id퐻 − id퐻1 ⊕ 0

= 0 ⊕ id퐻2

2 2 and thus 퐽2 = Π퐻2 (퐽Π퐻2 ) |퐻2 ∈ id퐻2 +풦(퐻2), showing that 퐽2 induces a polarization

풥2 of 퐻2. Now we want to prove that 풥 = 풥1 ⊕ 풥2. We have

퐽 − 퐽1 ⊕ 퐽2 = 퐽Π퐻1 − 퐽1 ⊕ 0 + 퐽Π퐻2 − 0 ⊕ 퐽2

≡ Π퐻2 퐽Π퐻2 − 0 ⊕ 퐽2 + Π퐻1 퐽Π퐻2

= Π퐻1 (퐽 − 퐽Π퐻1 ))

≡ Π퐻1 퐽 − 퐽1 ⊕ 0 ≡ 0

22 modulo 풦(퐻), proving [퐽] = [퐽1 ⊕ 퐽2].

We proceed to proving that (2) implies (1). From (2) we obtain that Π퐻2 [퐽, 퐽1 ⊕

0](퐽1 ⊕ 0) ≡ Π퐻2 퐽Π퐻1 is compact and

Π퐻1 [퐽, 퐽1 ⊕ 0]Π퐻1 ≡ Π퐻1 퐽Π퐻1 (퐽1 ⊕ 0) − (퐽1 ⊕ 0)Π퐻1 퐽Π퐻1

= [Π퐻1 퐽Π퐻1 , 퐽1 ⊕ 0]

is compact. The latter being compact tells us that [Π퐻1 퐽|퐻1 ] = [퐽1] and thus

Π퐻1 퐽Π퐻1 − 퐽1 ⊕ 0 is compact. It follows that

퐽Π퐻1 − 퐽1 ⊕ 0 = Π퐻2 퐽Π퐻1 + Π퐻1 퐽Π퐻1 ≡ 퐽1 ⊕ 0

which shows half of (1). However we also get that

2 (퐽1 ⊕ 0)[퐽, 퐽1 ⊕ 0] = (퐽1 ⊕ 0)퐽Π퐻1 (퐽1 ⊕ 0) − (퐽1 ⊕ 0) 퐽

≡ (퐽1 ⊕ 0)(퐽1 ⊕ 0)(퐽1 ⊕ 0) − Π퐻1 퐽

≡ −(Π퐻1 퐽 − 퐽1 ⊕ 0)

is compact, proving the other half of (1). Finally we prove that (1) implies (2). This follows immediately since

2 2 [퐽, 퐽1 ⊕ 0] = 퐽Π퐻1 (퐽1 ⊕ 0) − (퐽1 ⊕ 0)Π퐻1 퐽 ≡ (퐽1 ⊕ 0) − (퐽1 ⊕ 0) = 0.

Other natural operation on polarizations are pullbacks and pushforwards.

Proposition 2.20. Let 퐴 ∈ ℱ(퐻1, 퐻2), and 풥푖 be polarizations on 퐻푖 represented 2 by 퐽푖. Let 퐵 ∈ ℱ(퐻2, 퐻1) be any Fredholm inverse to 퐴. Then (퐵퐽2퐴) ∈ 1 + 풦(퐻1) 2 and (퐴퐽1퐵) ∈ 1 + 풦(퐻2).

2 Proof. (퐵퐽2퐴) = 퐵퐽2퐴퐵퐽2퐴 ≡ 퐵퐽2퐽2퐴 ≡ 퐵퐴 ≡ 1 modulo 풦(퐻1). The other identity follows similarly.

23 Since all Fredholm inverses of 퐴 differ by a compact operator we get well-defined polarizations:

Definition 2.21. Let 퐴 ∈ ℱ(퐻1, 퐻2). Choose any Fredholm inverse 퐵 ∈

ℱ(퐻2, 퐻1). * We define the pullback 퐴 풥2 = [퐵퐽2퐴] for any 퐽2 ∈ 풥2 representing a polarization on 퐻2, and the pushforward 퐴*풥1 = [퐴퐽1퐵] for any 퐽1 ∈ 풥1 representing a polarization on 퐻1.

* * Notice that 퐴*풥1 = 퐵 풥1 and 퐴 풥2 = 퐵*풥2 for 퐵 a Fredholm inverse to 퐴.

Proposition 2.22. For 퐴 ∈ ℱ(퐻1, 퐻2) and any Fredholm inverse 퐵 ∈ ℱ(퐻2, 퐻1) the following are equivalent:

1. 퐴 ∈ ℱ푟푒푠(퐻1, 퐻2),

* 2. 퐴 풥2 = 풥1,

3. 퐴*풥1 = 풥2,

4. 퐵 ∈ ℱ푟푒푠(퐻2, 퐻1).

Proof. Choose a Fredholm inverse 퐵 ∈ ℱ(퐻2, 퐻1) to 퐴.

Assume (1). Thus [퐵퐽2퐴, 퐽1] = 퐵퐽2퐴퐽1 −퐽1퐵퐽2퐴 ≡ 퐵퐴퐽1퐽1 −퐽1퐵퐴퐽1 ≡ id−id modulo compact operators, implying (2). But also [퐴퐽1퐵, 퐽2] = 퐴퐽1퐵퐽2 −퐽2퐴퐽1퐵 ≡

퐽2퐴퐵퐽2 − 퐽2퐽2퐴퐵 ≡ 0, implying (3). Conditions (1) and (4) are equivalent by Proposition 2.16.

Assume (2). Then 퐴퐽1 − 퐽2퐴 = 퐴퐽1 − 퐴(퐵퐽2퐴) ≡ 퐴퐽1 − 퐴퐽1 = 0, thus (1) follows.

Similarly (or using the previous result for 퐵), (3) implies (1).

The topologies of these sets of operators depend on the dimensionalities of 퐻±. If 퐻 is finite dimensional, all endomorphisms are both Fredholm and compact, thus

24 퐵(퐻) = ℱ(퐻) = ℱ푟푒푠(퐻) = 풦(퐻) ≃ * is contractible. On the other hand, GL(퐻) is the classical general linear group, having nontrivial topology, in particular two connected components. The situation is different when 퐻 is infinite dimensional. Let us first investigate the case when 퐻 is either positively or negatively oriented, but infinite dimensional.

Then we have 퐵(퐻) = 퐵푟푒푠(퐻) which is contractible, ℱ(퐻) = ℱ푟푒푠(퐻) which has the homotopy type of Z × 퐵푂, while GL(퐻) = GL푟푒푠(퐻) is contractible by Kuipers’ theorem. Note that the projection ℱ(퐻) → Z is given by taking the index of the map and induces isomorphism on 휋0. Finally, we investigate the case when 퐻 is not negatively nor positively polarized, i.e., it is properly polarized. Any proper polarization is induced by a splitting 퐻 = 퐻+ ⊕ 퐻− for which both 퐻± are infinite dimensional, i.e., 퐻 ≃ 퐻+ ≃ 퐻−. We fix such a splitting.

Proposition 2.23. The group GL푟푒푠(퐻) is homotopy equivalent to ℱ(퐻), which in turn is homotopy equivalent to Z × 퐵푂. the map GL푟푒푠(퐻) → ℱ(퐻) is given by ⎛ ⎞ 푎 푏 ⎜ ⎟ mapping ⎝ ⎠ ↦→ 푑, while the map GL푟푒푠(퐻) → Z is given by mapping 퐴 = 푐 푑 ⎛ ⎞ 푎 푏 ⎜ ⎟ ⎝ ⎠ ↦→ ind 푑 =: ind−(퐴) = − ind+(퐴) and induces an isomorphism on 휋0. 푐 푑 2 The group ℱ푟푒푠(퐻) is homotopy equivalent to (ℱ(퐻)) which in turn is homotopy ⎛ ⎞ 푎 푏 2 2 ⎜ ⎟ equivalent to Z × (퐵푂) , the first equivalence given by mapping ⎝ ⎠ ↦→ (푎, 푑) 푐 푑 ⎛ ⎞ 푎 푏 ⎜ ⎟ and the second one by mapping 퐴 = ⎝ ⎠ ↦→ (ind 푎, ind 푑) =: (ind+(퐴), ind−(퐴)) 푐 푑 and inducing an isomorphism on 휋0.

The numbers (ind+(퐴), ind−(퐴) are well-defined, i.e., do not depend on the decomposition 퐻 = 퐻+ ⊕ 퐻− inducing the chosen polarization.

Proof. The first result is proven in [PS86, Chapter 6] (also compare [CJS95] and

2 [Wur06]). The second result follows from Proposition 2.16 since ℱ푟푒푠(퐻) ≃ (ℱ(퐻)) × 2 (풦(퐻)) , the last factor being contractible.

25 To develop the theory of semi-infinite dimensional cycles we need to define whatwe mean for a linear map to a polarized space to be “aligned to” or “comparable with” the 퐻− component of a chosen splitting. We start by giving a concrete definition utilizing a splitting and only after that providing a more abstract equivalent.

Definition 2.24 (negative map). Let 퐻 = 퐻+ ⊕ 퐻− be a decomposition representing 풥 . A linear map 푠 : 푉 → 퐻 from a Hilbert space 푉 to 퐻 is called a negative (resp. positive) map if 푠+ = Π+푠 : 푉 → 퐻+ is compact (resp. Fredholm) and 푠− = Π−푠 : 푉 → 퐻− is Fredholm (resp. compact). The set of such maps is denoted

by ℱ−(푉, 퐻) (resp. ℱ+(푉, 퐻)).

Proposition 2.25. A map 푠 ∈ Hom(푉, 퐻) is negative (resp. positive) if and only if there exists a Hilbert space 푊 and a map 푡 ∈ Hom(푊, 퐻) such that 푡 + 푠 ∈

ℱ푟푒푠(푊 ⊕ 푉, 퐻) where the polarization on 푊 ⊕ 푉 is the one coming from the direct

sum, i.e., 퐽|푊 = id and 퐽|푉 = −id. Moreover, in this case the set of maps 푡 satisfying the above conditions is exactly the set of positive (resp. negative) maps to 퐻.

Proof. We will show this equivalence for negative 푠. Firstly, since all 퐽 ∈ 풥 are congruent modulo compact operators, the choice of 퐽 does not matter at all. On one hand, if 푠 is negative, then one can take the inclusion 푡 : 퐻+ ˓→ 퐻 to satisfy all of the requirements, which follows from Proposition 2.16. On the other hand, having such 푡 we may write 푡 + 푠 : 푊 ⊕ 푉 → 퐻+ ⊕ 퐻− ⎛ ⎞ 푎 푏 2 ⎜ ⎟ (after having chosen 퐽 such that 퐽 = 1) in block decomposition 푡 + 푠 = ⎝ ⎠. 푐 푑 − 1−퐽 + 1+퐽 Now notice that 푏 = Π 푡 = 2 푡 and 푐 = Π 푠 = 2 푠 are both compact, and thus Proposition 2.16 implies Π−푠 = 푑 : 푉 → 퐻− is Fredholm. From this block decomposition it follows that for a negative map 푠, 푡 satisfies the

required conditions if and only if it is positive.

26 Fixing a decomposition 퐻 = 퐻+ ⊕퐻− we can define the index of 푠. For a negative

− + (resp. positive) map 푠 we could take ind−(푠) = ind(Π 푠) (resp. ind+(푠) = ind(Π 푠)).

Such index is independent of deformations of 푠, i.e., if {푠푡}푡∈[푎,푏] is a continuous family ′ of maps 푠푡 : 푉 → 퐻, then ind−(푠푡) = ind−(푠푡′ ) for any 푡, 푡 ∈ [푎, 푏]. However, it does depend on the choice of the decomposition 퐻 = 퐻+ ⊕ 퐻−. What turns out to be independent is the difference of indices between two such 푠.

Definition 2.26 (relative index of negative (positive) maps). Let

푠푖 ∈ Hom(푉푖, 퐻) be negative (resp. positive) maps. Take any positive (resp.

negative) map 푡 ∈ Hom(푊, 퐻) and define ind(푠1, 푠2) = ind−(푠1 + 푡) − ind−(푠2 + 푡)

(resp. ind(푠1, 푠2) = ind+(푠1 + 푡) − ind+(푠2 + 푡)).

Proposition 2.27. The definition of relative index is independent of the choice of 푡

and of the deformations of 푠푖, satisfies a cocycle condition (for 푠1, 푠2, 푠3 be negative

(resp. positive) maps to 퐻 one has ind(푠1, 푠2)+ind(푠2, 푠3) = ind(푠1, 푠3)) and satisfies

ind(푠1, 푠2) = 0 whenever 푠2 is a deformation of 푠1. Moreover, for any choice of a decomposition 퐻 = 퐻+ ⊕ 퐻− representing the

polarization we have ind(푠1, 푠2) = ind−(푠1) − ind−(푠2) (resp. ind(푠1, 푠2) = ind+(푠1) −

ind+(푠2)).

Proof. Say that 푠푖 are negative. Take any positive 푡 and any decomposition 퐻 = + − 퐻 ⊕ 퐻 . We need to prove that ind(푠1 + 푡) − ind(푠2 + 푡) = ind−(푠1) − ind−(푠2). ⎛ ⎞ 푎 푏푖 ⎜ ⎟ + − Indeed, we have 푡 + 푠푖 = ⎝ ⎠ : 푊 ⊕ 푉푖 → 퐻 ⊕ 퐻 . Proposition 2.16 implies 푐 푑푖 that ind(푡 + 푠푖) = ind(푎) + ind(푑푖) = ind(푎) + ind−(푠푖) and the result follows.

We would like to be able to talk about the index of a negative map as well as the index of a correspondence (which will be defined soon). For that, we could either require to choose specific (equivalence classes of) decompositions inducing the polarizations and obtain indices that are integers. For future applications, however, we want to be able to talk about indices without making such choices.

27 Definition 2.28 (index set). Let 퐻 be a polarized Hilbert space and assume that 퐻 is not positively polarized. We denote by 퐼(퐻) and call the index set the set of

equivalence classes of negative maps to 퐻, where 푠1 ∼ 푠2 if and only if ind(푠1, 푠2) = 0. Whenever 푠 is a negative map to 퐻, we denote by ind 푠 its equivalence class [푠] ∈ 퐼(퐻).

Whenever [푠] ∈ 퐼(퐻) and 푘 ∈ Z, we define [푠]+푘 for 푠 : 푉 → 퐻 the following way. 푘 If 푘 ≥ 0 and , then [푠]+푘 is defined to be [푠+0푘]: 푉 ⊕R → 퐻, [푠+0푘](푣, 푤) = 푣. If 푘 < 0, then we choose any codimension-(−푘) subspace 푊 ⊂ 푉 and define [푠] + 푘 =

[푠|푊 ]. For 퐻 positively polarized, we define 퐼(퐻) = Z with the canonical action of Z by addition and define [푠] = ind 푠 = dim 푉 for 푠 : 푉 → 퐻.

Lemma 2.29. There is a Z-equivariant bijection 퐼(퐻) ≃ Z.

Proof. Choose a decomposition 퐻 = 퐻+ ⊕ 퐻− inducing the polarization. The isomorphism is given by mapping [푠] to ind− 푠.

Equivariance follows from the fact that [푠2] = [푠1] + ind(푠1, 푠2).

The proof shows that a decomposition 퐻 = 퐻+ ⊕퐻− ging a distinguished element

of the index set [퐻−] ∈ 퐼(퐻) provides an isomorphism 휑 : 퐼(퐻) → Z of Z-torsors,

such that 휑([푠]) = ind− 푠. This shows that to define the negative index as an element of integers one needs only to choose the isomorphism 휑, not a specific decomposition. We formalize this viewpoint.

Definition 2.30 (graded polarization). A(Z/(푑)-)grading on a polarized space ℋ is a Z-equivariant map 휑 : 퐼(퐻) → Z/(푑) for some 푑 ≥ 0. We call the pair 풥˜ = (풥 , 휑) a (Z/(푑))-graded polarization.

We (re)define the negative index using this identification via ind−(푠) = 휑([푠]).

28 Lemma 2.31. Let 휑푖 : 퐼(퐻) → Z/(푑) be two gradings coming from decompositions + − 퐻 = 퐻푖 ⊕ 퐻푖 inducing the same polarization. Then the grading difference 휑2 − 휑1 = − − ind(휄1, 휄2) = ind(Π | − ) where 휄푖 : 퐻 → 퐻 denote the inclusion maps. 2 퐻1 푖

Moreover, if 퐽푖 ∈ 풥 are induced by these polarizations, then 휑2 − 휑1 is the spectral flow (modulo 푑) from 퐽1 to 퐽2.

Proof. By Proposition 2.27 we get 휑2 −휑1 = 휑2(휄1)−휑1(휄1) = ind−,2(휄1)−ind−,1(휄1) = − ind−,2(휄1)−0 = ind−,2(휄1)−ind−,2(휄2) = ind(휄1, 휄2). Moreover, ind−,2(휄1) = ind(Π | − ) 2 퐻1 by the previous definition of negative index, as wished.

± The last statement follows from the fact that 퐻푖 are the ±1-eigenspaces of 퐽푖.

The action of Z constructed so far is very concrete but there is a more natural way of understanding it.

Definition 2.32 (action of operators on index sets). Assume 퐻1 is not positively polarized. Any operator 퐴 ∈ ℱ푟푒푠(퐻1, 퐻2) induces a map 퐼(퐴): 퐼(퐻1) →

퐼(퐻2) given by [푠] ↦→ [퐴 ∘ 푠].

Assume 퐻1 is positively polarized and 퐴 ∈ ℱ푟푒푠(퐻1, 퐻2) (thus 퐻2 is also positively polarized). Then the map 퐼(퐴): 퐼(퐻1) → 퐼(퐻2) is defined to be id : Z → Z

(which restricts to [푠] ↦→ [퐴 ∘ 푠] on Z≥0.)

Lemma 2.33. 퐼(퐴) is well-defined, functorial in 퐴, stable under deformations of 퐴

(i.e., it descends to 휋0(ℱ푟푒푠(퐻1, 퐻2))) and Z-equivariant. ˜ If 풥푖 are graded polarizaions with respective gradings 휑푖 : 퐼(퐻푖) → Z/(푑), then −1 ind−(퐴) = 휑2∘퐼(퐴)∘휑1 is well-defined as an element of Z/(푑) ≃ MorZ(Z/(푑), Z/(푑)).

Moreover, ind−(퐴 ∘ 푠) = ind−(푠) + ind−(퐴) for any negative subspace 푠 and ind−(퐵 ∘ ˜ 퐴) = ind−(퐴) + ind−(퐵) for any 퐵 ∈ ℱ푟푒푠(퐻2, 퐻3) (given a graded polarization 풥3).

Proof. It is well-defined; indeed, for any choice of positive map 푡 : 푊 → 퐻1 we have that ind(퐴 ∘ 푠1, 퐴 ∘ 푠2) = ind(퐴 ∘ (푠1 + 푡)) − ind(퐴 ∘ (푠2 + 푡)) = ind 퐴 + ind(푠1 + 푡) − ind 퐴 − ind(푠2 + 푡) = ind(푠1, 푠2).

29 Functoriality and stability under continuous deformations of 퐴 follow directly from the definition and stability of Fredholm index under continuous deformations.

Finally, the equivariance follows from the fact that [퐴 ∘ (푠 + 0푘)] = [(퐴 ∘ 푠) + 0푘] = 푘 [퐴 ∘ 푠] + 푘 for any 푘 and the zero map 0푘 : R → 퐻. The results about negative indices follow from the fact that ind(퐺∘퐹 ) = ind(퐹 )+ ind(퐺) for any Fredholm operators 퐹, 퐺.

+ − Choose decompositions 퐻푖 = 퐻푖 ⊕ 퐻푖 . In such case we get ind−(퐴 ∘ 푠) = ± ± ind−(퐴) + ind−(푠). In particular, if 퐻1 = 퐻2 and 퐻1 = 퐻2 then we obtain ind−(퐴 ∘

푠) − ind−(푠) = ind−(퐴) = − ind+(퐴) ∈ Z. The right hand side is independent of the choice of a specific decomposition. If we further assume that 퐴 ∈ GL푟푒푠(퐻) then functoriality gives an action of Z ≃ 휋0(GL푟푒푠(퐻)) on 퐼(퐻).

Lemma 2.34. The 휋0(GL푟푒푠(퐻)) ≃ind− Z action coincides with the Z action on 퐼(퐻) defined in Definition 2.28.

Proof. By transitivity of these actions we just need to prove that some element 퐴 ∈

GL푟푒푠(퐻) which has ind−(퐴) = − ind+(퐴) = 1 acts on a chosen negative map 푠 : − 퐻 → 퐻 the same way as 1 would act on 푠 as defined earlier. Take 푠 = id|퐻− ⎛ ⎞ 푎 푏 ⎜ ⎟ + + and any 퐴 = ⎝ ⎠ such that 푎 : 퐻 → 퐻 is injective with ind 푎 = −1 and 0 푑 − − 푑 : 퐻 → 퐻 is given in some orthonormal coordinates by (푎1, 푎2,...) ↦→ (푎2, 푎3,...). − − Thus 퐴 ∘ 푠 = 푑 ∘ 푠 : 퐻 → 퐻 is isomorphic to 0 + 푠 : R ⊕ 퐻 → 퐻, as wished.

2.3 Linear correspondences

We complete the discussion of polarized Hilbert spaces by introducing and analyzing correspondences between them.

+ − Definition 2.35 (linear correspondence). Let 퐻푖 = 퐻푖 ⊕ 퐻푖 represent 풥푖 for

푖 = 0, 1.A (linear) correspondence from 퐻0 to 퐻1 is a Hilbert space 푍 and a con- 푠 푡 tinuous linear map 푓 : 푍 → 퐻0 ⊕ 퐻1 (which we also denote as a pair (푓 , 푓 )) such that either of the following two axioms holds:

30 + 푠 − 푡 + − + 푡 Axiom L1. (Π 푓 , Π 푓 ): 푍 → 퐻0 ⊕ 퐻1 is Fredholm and Π 푓 |ker(Π+푓 푠) : + 푠 + ker(Π 푓 ) → 퐻1 is compact.

+ 푠 − 푡 + − Axiom L1’. (Π 푓 , Π 푓 ): 푍 → 퐻0 ⊕ 퐻1 is Fredholm. If {푣푖} ⊂ 푍 is bounded and + 푠 + 푡 {Π 푓 (푣푖)} is precompact, then {Π 푓 (푣푖)} is precompact.

푠 We call 푓 a dense correspondence if additionally 푓 : 푍 → 퐻0 is dense. For the

chosen decompositions, we define the negative index of 푓 to be equal to ind− 푓 := ind((Π+푓 푠, Π−푓 푡)).

Remark 2.36. We use the notations 푓 푠 and 푓 푡 to distinguish between the source and target maps, two components of 푓. Interchanging the two does not, in general, give a correspondence from 퐻1 to 퐻0, even though many correspondences in this article will satisfy this property.

Lemma 2.37. For a continuous linear map 푓 : 푍 → 퐻0 ⊕ 퐻1 and fixed decomposi- + − tions 퐻푖 = 퐻푖 ⊕ 퐻푖 Axioms L1 and L1’ are equivalent.

Proof. Axiom L1’ directly imples Axiom L1.

+ 푠 Assume Axiom L1 holds. Let {푣푖} ⊂ 푍 be bounded and {Π 푓 (푣푖)} be precom- + 푠 − 푡 + 푠 + pact. Since (Π 푓 , Π 푓 ) is Fredholm, therefore Π 푓 : 푍 → 퐻0 has closed image, coker(Π+푓 푠) is finite-dimensional and there is a continuous map 푙 : im(Π+푓 푠) → 푍

+ 푠 + 푠 + 푠 such that Π 푓 ∘ 푙 = Idim Π+ . Take 푤푖 = 푙(Π 푓 (푣푖)). Then {푤푖} = 푙({Π 푓 (푣푖)}) + 푠 + 푠 + 푠 is precompact and Π 푓 (푤푖) − Π 푓 (푣푖) = 0, i.e., 푧푖 = 푤푖 − 푣푖 ∈ ker Π 푓 . But + 푡 + 푡 푧푖 is also bounded, so by compactness of Π 푓 |ker Π+푓 푠 we obtain that {Π 푓 (푧푖)} is + 푡 + 푡 + 푡 precompact. It follows that {Π 푓 (푣푖)} = {Π 푓 (푤푖) + Π 푓 (푧푖)} is precompact.

Lemma 2.38. The definition of a linear correspondence depends only on the choice

+ − of polarizations 풥푖 but not on the choice of decompositions 퐻푖 = 퐻푖 ⊕퐻푖 representing them.

Proof. For two chosen decompositions the corresponding Axioms L1’ are equivalent

± ± since the difference of projections Π2 − Π1 are compact.

31 Lemma 2.39. A map 푠 : 푉 → 퐻 is a negative map if and only if it is a linear correspondence from {0} to 퐻, and its negative index as a negative map is equal to its index as a correspondence.

The main result of this section, justifying these definitions, is that a composition of such correspondences is itself a correspondence.

Proposition 2.40 (composition of linear correspondences). Let 푓 : 푍 → 퐻0⊕

퐻1 and 푔 : 푊 → 퐻1 ⊕ 퐻2 be correspondences of polarized Hilbert spaces. Then 푔 ∘ 푓,

defined as the fiber sum 푓 ⊕퐻1 푔 : 푍 ⊕퐻1 푊 → 퐻0 ⊕ 퐻2 is a correspondence from 퐻0 푡 푠 to 퐻2 and ind−(푔 ∘ 푓) = ind− 푓 + ind− 푔 − dim coker(푓 − 푔 ).

If 푔 is a dense correspondence, then for fixed decompositions of 퐻푖 we have ind−(푔∘

푓) = ind− 푓 + ind− 푔. Moreover, if both 푓 and 푔 are dense, then so is 푔 ∘ 푓.

In the course of the proof of the proposition, we will use a slightly non-standard description of Fredholmness, utilizing the following lemma.

Lemma 2.41. Let 퐹 : 퐴 → 퐵 be a continuous linear map of Hilbert spaces. Then:

• If 퐹 has finite dimensional kernel and cokernel (as vector spaces), then itis Fredholm (i.e., it follows that 퐹 (퐴) is closed).

• 퐹 has finite dimensional kernel if and only if any bounded sequence {푎푖} ⊂ ker 퐹 is precompact.

• 퐹 has finite dimensional cokernel if and only if any for any bounded sequence

{푏푖} ⊂ 퐵 there exists a bounded sequence {푎푖} ⊂ 퐴 such that {푏푖 − 퐹 (푎푖)} ⊂ 퐵 is precompact.

˜ ˜ Proposition 2.42. Let (퐹, 퐺): 퐻 → 퐴 ⊕ 퐵 be Fredholm. Then 퐹 = 퐹 |ker 퐺 : 퐻 = ker 퐺 → 퐴 is Fredholm and has index equal to ind((퐹, 퐺)) − dim coker 퐺.

Proof. This follows from applying Proposition 2.43, with 퐶 = ker 퐺, 퐷 ⊂ 퐻 any closed subspace complimentary to 퐶, 푓 ⊕ ℎ = 퐹 and 푔 = 퐺|퐷.

32 ⎛ ⎞ 푓 ℎ ⎜ ⎟ Proposition 2.43. Let 푀 = ⎝ ⎠ : 퐶 ⊕ 퐷 → 퐴 ⊕ 퐵 be Fredholm and let 0 푔 dim ker 푔 < ∞. Then 푓, 푔 are both Fredholm and ind 푓 + ind 푔 = ind 푀. Conversely, if 푓, 푔 are Fredholm, then 푀 is Fredholm, too. ⎛ ⎞ 푘 푙 ⎜ ⎟ Proof. Let 푁 = ⎝ ⎠ be a Fredholm inverse to 푀, i.e., 푁푀 − 퐼 and 푀푁 − 퐼 are 푚 푛 compact. It follows that 푔푛 − 퐼 is compact, which implies that the 푔푛 is Fredholm, thus the cokernel of 푔 is finite-dimensional. By Lemma 2.41, 푔 is Fredholm. We also have that 푔푚 is compact, but since 푔 is Fredholm, it implies that 푚 is compact. Finally, since 푘푓 − 퐼 and 푓푘 + ℎ푚 − 퐼 are compact, therefore also 푓푘 − 퐼 is compact and it follows that 푓 is Fredholm, too. We proceed to computing the index of 푀. Let 푓,˜ 푔˜ be any Fredholm inverses of

푓, 푔, respectively. Take a continuous family of operators ℎ푡 such that ℎ0 = 0, ℎ1 = ℎ. ˜ ˜ Define ℎ푡 = −푓ℎ푡푔˜. Notice

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ˜ ˜ ˜ ˜ ˜ 푓 ℎ푡 푓 ℎ푡 푓푓 푓ℎ푡 − 푓ℎ푡푔푔˜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = ⎝ ⎠ 0푔 ˜ 0 푔 0푔푔 ˜

˜ ˜ and since 푔푔˜ = 1 + 퐾 for some compact 퐾, it follows that 푓ℎ푡 − 푓ℎ푡푔푔˜ is compact; ⎛ ⎞ 푓 ℎ푡 ⎜ ⎟ thus, the operator 푀푡 = ⎝ ⎠ is left Fredholm. Moreover, 0 푔

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ˜ ˜ ˜ ˜ 푓 ℎ푡 푓 ℎ푡 푓푓 −푓푓ℎ푡푔˜ + 푔푔˜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = ⎝ ⎠ 0 푔 0푔 ˜ 0 푔푔˜

and again the upper right entry of this matrix is compact, so 푀푡 is actually a family of Fredholm operators. Therefore ind 푀 = ind 푀1 = ind 푀0 = ind diag(푓, 푔) = ind 푓 + ind 푔, as wished. Note that this argument also shows that if 푓, 푔 are Fredholm, then 푀 is Fredholm, which completes the proof.

Proof of Proposition 2.40. First of all, notice that 푍 ⊕퐻1 푊 is a closed linear subspace

33 + − of 푍 ⊕푊 , so it is a Hilbert space. Choose decompositions 퐻푖 = 퐻푖 ⊕퐻푖 representing

the polarizations 풥푖.

+ 푠 − 푡 Axiom L1. Firstly, we want to show that (Π (푔 ∘ 푓) , Π (푔 ∘ 푓) ): 푍1 ⊕퐻1 푍2 → + − + 푠 − 푡 퐻0 ⊕ 퐻2 is Fredholm. By Proposition 2.42, it suffices to prove that (Π 푓 , Π 푓 − − 푠 + 푡 + 푠 − 푡 + − + − Π 푔 , −Π 푓 + Π 푔 , Π 푔 ): 푍 ⊕ 푊 → 퐻0 ⊕ 퐻1 ⊕ 퐻1 ⊕ 퐻2 is Fredholm. + − + + 푡 ˜ ˜ Let 퐴 : 퐻0 ⊕ 퐻1 → 퐻1 be given by 퐴 = Π 푓 푓, where 푓 is a Fredholm inverse to (Π+푓 푠, Π−푓 푡). Note that since 푓˜∘ (Π+푓 푠, Π−푓 푡) − Id is compact, therefore (after

+ 푡 + 푡 + 푠 − 푡 + composing with Π 푓 ) we get that Π 푓 − 퐴(Π 푓 ⊕ Π 푓 ): 푍 → 퐻1 is compact. + 푡 − 푡 + 푠 + It also follows that Π 푓 |ker Π+푓 푠 − 퐴(0 ⊕ Π 푓 ) : ker Π 푓 → 퐻1 is compact, so − 푡 + 푠 + 퐴(0 ⊕ Π 푓 ) : ker Π 푓 → 퐻1 is compact. However, by Proposition 2.42, the map − 푡 ˜ − 푡 − + 푠 Π 푓 |ker Π+푓 푠 is Fredholm. Take its Fredholm inverse Π 푓 : 퐻1 → ker Π 푓 . Then − 푡 − 푡 퐴(0 ⊕ Π 푓 )Π˜ 푓 is compact but it also differs from 퐴| − by a compact map, and 퐻1 thus 퐴| − is compact. 퐻1 Recall that we are considering the map 퐹 = (Π+푓 푠, Π−푓 푡 − Π−푔푠, −Π+푓 푡 +

+ 푠 + 푡 + − + − Π 푔 , Π 푔 ): 푍 ⊕ 푊 → 퐻0 ⊕ 퐻1 ⊕ 퐻1 ⊕ 퐻2 and we want to prove it is Fred- + − + − + − + − holm. Consider the map 퐵 : 퐻0 ⊕ 퐻1 ⊕ 퐻1 ⊕ 퐻2 → 퐻0 ⊕ 퐻1 ⊕ 퐻1 ⊕ 퐻2 given by 퐵(푎, 푏, 푐, 푑) = (푎, 푏, 푐 + 퐴(푎 ⊕ 푏), 푑). Note that this is invertible. Composing it with 퐹

+ 푠 − 푡 − 푠 + 푡 + 푠 − 푡 − 푠 we get 퐹˜ = 퐵 ∘ 퐹 = (Π 푓 , Π 푓 − Π 푔 , −Π 푓 + 퐴(Π 푓 ⊕ Π 푓 ) − 퐴| − Π 푔 + 퐻1 + 푠 − 푡 + 푡 + 푠 − 푡 Π 푔 , Π 푔 ). We have proven above that both −Π 푓 +퐴(Π 푓 ⊕Π 푓 ) and 퐴| − are 퐻1 compact, and thus 퐹˜ differs by a compact map from (Π+푓 푠, Π−푓 푡−Π−푔푠, Π+푔푠, Π−푔푡). This last map is Fredholm by Proposition 2.43 and has index equal to ind 푓 + ind 푔, and 퐹 has the same index. Proposition 2.42 implies that ind−(푔 ∘ 푓) = ind− 푓 + 푡 푠 ind− 푔 − dim coker(푓 − 푔 ), as wished.

+ 푠 We also need to prove that when {(푧푖, 푤푖)} ⊂ 푍 ⊕퐻1 푊 is bounded and {Π 푓 (푧푖)} + 푡 is precompact, then {Π 푔 (푤푖)} is precompact (since Axiom L1 is equivalent to Axiom + 푡 + 푠 L1’). But since 푓, 푔 satisfy Axiom L1’, this implies that {Π 푓 (푧푖) = Π 푔 (푤푖)} is + 푡 precompact and therefore {Π 푔 (푤푖)} is precompact, too. Denseness. Assume that 푓 and 푔 are dense. We want to prove that 푔 ∘ 푓 is

푠 dense. Arguing by contradiction, assume that the image of (푔 ∘ 푓) : 푍 ⊕퐻1 푊 → 퐻0 푠 is not dense in 퐻0. Thus, we can choose 푣 ∈ 퐻0 such that 푣 ̸= 0 and 푣 ⊥ im(푔 ∘ 푓) .

34 푠 Choose any negative map 푠 : 푃 → 퐻0 such that im 푠 ⊥ 푣. This implies 푠 − (푔 ∘ 푓) :

푃 ⊕ (푍 ⊕퐻1 푊 ) → 퐻0 has image perpendicular to 푣, thus is not onto. Also, by the proof above we already know that 푓 ∘ 푠 is a negative map in 퐻1.

푠 − 푡 − Furthermore, (푠 − 푓 , Π 푓 ): 푃 ⊕ 푍 → 퐻0 ⊕ 퐻1 is Fredholm (by the proof of 푠 Axiom L1 above) and thus 푠 − 푓 : 푃 ⊕ 푍 → 퐻0 has finite dimensional cokernel. On 푠 푠 the other hand, im 푓 is dense in 퐻0, so the image of 푠 − 푓 is dense, too, and thus 푠 푠 푠 − 푓 is onto 퐻0. The same argument shows that 푓 ∘ 푠 − 푔 :(푃 ⊕퐻1 푍) ⊕ 푊 → 퐻1 is onto.

We want to prove that 푃 ⊕ (푍 ⊕퐻1 푊 ) → 퐻0 is onto. Take 푎 ∈ 퐻0. There is (˜푝, 푧˜) ∈ 푃 ⊕푍 mapping to 푎 under 푠−푓 푠, i.e., 푎+푓 푠(˜푧) = 푠(˜푝). There is also (¯푝, 푧,¯ 푤¯) ∈

푡 푠 푡 푠 푡 (푃 ⊕퐻0 푍)⊕푊 mapping to −푓 (˜푧) ∈ 퐻1 under 푓∘푠−푔 , i.e., 푓 (¯푧)−푔 (푤 ¯) = −푓 (˜푧) and 푠(¯푝) = 푓 푠(¯푧). Finally, the triple (푝 =푝 ˜+푝, ¯ 푧 =푧 ˜+푧, ¯ 푤 =푤 ¯) satisfies 푓 푡(푧) − 푔푠(푤) = 푓 푡(¯푧) − 푔푠(푤 ¯) + 푓 푡(˜푧) = 0 and 푠(푝) − 푓 푠(푧) − 푎 = 푠(˜푝) − 푓 푠(˜푧) − 푎 + 푠(¯푝) − 푓 푠(¯푧) = 0,

푠 which means (푝, 푧, 푤) ∈ 푃 ⊕ (푍 ⊕퐻1 푊 ) and it maps to 푎 under (푔 ∘ 푓) . This contradicts the earlier statement that the image of 푠 − (푔 ∘ 푓)푠 does not contain 푎 ̸= 0 and finishes the proof by contradiction.

Corollary 2.44. Let 푠 : 푃 → 퐻0 be a negative map of 퐻0 and 푓 : 푍 → 퐻0 ⊕ 퐻1 be a correspondence from 퐻0 to 퐻1. Then 푓 ∘ 푠 := 푠 ⊕퐻0 푓 : 푃 ⊕퐻0 푍 → 퐻1 is a negative map.

Moreover, for fixed decompositions we have ind−(푓 ∘ 푠) = ind− 푠 + ind− 푓 − 푠 dim coker(푠 − 푓 ), and if 푓 is dense, then ind−(푓 ∘ 푠) = ind− 푠 + ind− 푓.

Corollary 2.45. Any correspondence 푓 : 푍 → 퐻0 ⊕ 퐻1 induces a Z-equivariant map

퐼(푓): 퐼(퐻0) → 퐼(퐻1) defined by [푠] ↦→ [푓 ∘ 푠] whenever 푠 t 푓. 퐼(푓) is stable under continuous deformations of 푓 and functorial in 푓. Moreover,

−1 ind−(푓) = 휑1 ∘ 퐼(푓) ∘ 휑0 and thus depends only on the choices of (Z/(푑)-)graded ˜ polarizations 풥푖 and not on the specific decompositions inducing them.

− Proof. We can always find a negative map 푠 : 푉 → 퐻0 such that 푠 t 푓 and 푓 ∘ 푠 is nonempty. This defines 퐼(푓)([푠]) = [푓 ∘푠] and we extend this map via Z-equivariance.

35 Functoriality follows from Proposition 2.40, and the relation with graded polar-

izations follows from the definition of 퐼(푓).

Remark 2.46. If 푓 is not dense, taking a representative 푠 of a given class 푖 ∈ 퐼(퐻0) such that 푠 t 푓 may not be possible. After all, 0 ∈ im 푠 ∩ im 푓 always, even if for index reasons codim(im 푠 + im 푓) > 0 for any representative 푠 of 푖. However, we are able to find such representatives for all sufficiently “large” elements of 퐼(퐻0).

2.4 Polarized manifolds

We have dealt with polarized vector spaces in Sections 2.2 and 2.3. Here we use these notions to define polarized manifolds and polarized maps between them by introducing polarizations on their tangent spaces.

Definition 2.47 (polarized manifolds). Let 푀 be Hilbert manifold. A polarization of 푀 is a choice of a set 풥 of bundle endomorphism 퐽 : 푇 푀 → 푇 푀 congruent modulo 풦(푇 푀) such that 퐽 2 ∈ 1 + 풦(푇 푀), where 풦(푇 푀) ⊂ 퐵(푇 푀) is the subset of fiberwise compact endomorphisms. We call the polarization 풥 ∨ = −풥 the dual of 풥 .

In particular, a polarization of 푀 induces a polarization on each fiber 푇푚푀 which

we denote by 풥 |푚. The polarizations of 푀 are continuous choices of polarizations on the fibers. One could require the polarization tobe integrable by choosing of a polarization of the Hilbert space 퐻 on which 푀 is modelled and requiring the

charts to preserve the polarization, i.e., asking that 퐷휙푢 ∈ ℱ푟푒푠(푇푢푈, 푇휙(푢)퐻) for any chart 휙 : 푈 → 퐻 and point 푢 ∈ 푈. However, we will not need that. On the other hand, assuming 푀 is connected, we can interpret choosing a polarization of 푀 modelled on a Hilbert space 퐻 as a reduction of its structure group to

GL푟푒푠(퐻) for a chosen polarization of 퐻.

36 Here it is important that there are three kinds of polarizations of 퐻, and thus of 푀, which we allowed. There are the positive and negative polarizations, for which

simply GL푟푒푠(퐻) = GL(퐻). If a fiber 푇푚푀 is polarized positively or negatively, then ′ the operator 퐽푚 − (±id) is compact and therefore all operators 퐽푚′ − (±id) for 푚 in a neighborhood 푈 of 푚 are compact, so 푀 is polarized positively on negatively in 푈 (c.f. proof of Proposition 2.53). If 푀 is connected, then the whole manifold is thus polarized positively or negatively. The second case is when the polarization is induced by a decomposition 퐻 = 퐻+ ⊕ 퐻− for 퐻+ ≃ 퐻− ≃ 퐻 (since we work with separable spaces, all infinite-dimensional Hilbert spaces are isomorphic) and for any other decomposition 퐻 = 퐻˜ + ⊕ 퐻˜ − there * + − ˜ + ˜ − exists 푔 ∈ 퐺퐿푟푒푠(퐻) such that 푔 (퐻 ⊕퐻 ) = 퐻 ⊕퐻 . So, without losing generality, we may fix a decomposition (and thus a polarization) of 퐻 once and for all. We are now going to show that every polarization of 푀 induces a reduction of the structure group of 푀 to 퐺퐿푟푒푠(퐻) when 푀 is not positively or negatively polarized. Recall that the frame bundle of 푀 is the principal GL(퐻)-bundle 퐹 푀 → 푀 which fiber over 푚 ∈ 푀 is given by GL(퐻, 푇푚푀), and the action of 푔 ∈ GL(퐻) on 퐴 ∈ −1 GL(퐻, 푇푚푀) is given by the composition 푔(퐴) = 퐴 ∘ 푔 . There is a principal subbundle 퐹푟푒푠푀 → 푀 with fiber over 푚 ∈ 푀 given by GL푟푒푠(퐻, 푇푚푀) and the action of 푔 ∈ GL푟푒푠(퐻) given as before. This is the desired reduction of the structure group of 푀. It will follow from Proposition 2.53 that this is a bundle, i.e., it is locally trivializable.

On the other hand, having such a reduction 퐹푟푒푠푀 ⊂ 퐹 푀 → 푀 we can locally choose a section 퐴푈 : 푈 → 퐹푟푒푠푀|푈 and define polarization on 푈 by taking −1 a representative 퐽푈 = 퐴푈 퐽퐴푈 . It is straightforward to check that for two choices ′ −1 ′ ′−1 퐴푈 , 퐴푈 : 푈 → 퐹푟푒푠푀|푈 we have 퐴푈 퐽퐴푈 − 퐴푈 퐽퐴푈 fiberwise compact and therefore for any two open sets 푈, 푉 the difference 퐽푈 |푈∩푉 − 퐽푉 |푈∩푉 is fiberwise compact, so we can construct 퐽 on 푀 by using a partition of unity subordinate to some locally finite covering of 푀 ([Lan93, Chapter XXII, Theorem 4.1]).

Proposition 2.48. Choosing a polarization is equivalent to reducing the structure group of 푇 푀 from GL(퐻) to GL푟푒푠(퐻).

37 Example 2.49. Any finite-dimensional manifold 푀 admits a unique polarization

풥0 = [id] = [−id].

Example 2.50. Any Hilbert manifold 푀 admits a positive and a negative polariza-

tion, 풥± = [±id]. If 푀 is infinite-dimensional, these polarizations are not equivalent.

Example 2.51. Any linear polarization 풥 of a Hilbert space 퐻 induces a polarization of the Hilbert manifold 퐻.

Lemma 2.13 tells us that on each fibre 푇푚푀 we can choose a representative 2 + − 푗 ∈ 풥푚 such that 푗 = 1 and thus obtain a decomposition 푇푚푀 = 푇푚 푀 ⊕ 푇푚 푀. The converse is clearly true:

Lemma 2.52. A global decomposition 푇 푀 = 푇 +푀 ⊕ 푇 −푀 induces a polarization

풥 = 퐽 + 풦(푇 푀), where 퐽|푇 +푀 = id and 퐽|푇 −푀 = −id.

However, this time not every polarization of 푀 can be constructed this way. It is possible to do that locally.

Proposition 2.53. For any point 푚 ∈ 푇 푀 there is an open neighborhood 푈 and

+,푈 −,푈 a trivializable bundle decomposition 푇 푀|푈 = 푇 푀 ⊕ 푇 푀 inducing the polarization 풥 .

Proof. We proceed along the lines of proof of Lemma 2.13. Choose 퐽 : 푇 푀 → 푇 푀

representing 풥 . For 퐽푚 choose 0 < 푟 < 1 such that the intersection 푆−1(푟) ∩ 휎(퐽푚)

is empty, where 훾푟 = 푆−1(푟) ⊂ C is a circle of radius 푟 around −1, oriented coun-

terclockwise. Since the spectrum 휎(퐽푥) depends continuously on 푥 there is an open

neighborhood 푈 of 푚 such that for any 푥 ∈ 푈 the intersection 훾푟 ∩ 휎(퐽푥) is empty. We obtain the Riesz projections

∫︁ −,푟 1 1 Π푥 = 푑푧 : 푇푥푀 → 푇푥푀 2휋푖 훾푟 푧 − 퐽푥

−,푟 varying continuously in 푥, i.e., Π ∈ 퐵(푇 푀|푈 , 푇 푀|푈 ). As in Lemma 2.13 we define +,푟 −,푟 ˜ +,푟 −,푟 ˜ Π = 1 − Π , 퐽푟 = Π − Π and obtain 퐽 − 퐽푟 ∈ 풦(푇 푀|푈 ).

38 We denote 푇 ±,푟푀 = im Π±,푟. It remains to prove that 푈 can be chosen small enough for 푇 ±,푟푀 to be trivializ-

able on 푈. We can shrink 푈 to obtain a trivialization Φ: 푈 × 푇푚푀 → 푇 푀|푈 . Then ±,푟 ±,푟 ±,푟 Π Φ| ±,푟 : 푈 × 푇 푀 → 푇 푀| restricts to the identity map on {푚} × 푇 푀. 푇푚 푀 푚 푈 푚 ±,푟 This means we can further shrink 푈 so that any restriction to {푢} × 푇푚 푀 is an iso- ±,푟 morphism onto image. But the image is exactly 푇푢 푀, which finishes the proof.

Corollary 2.54. There is a locally finite covering 풰 of 푀 by open sets 푈 ∈ 풰 to-

+,푈 −,푈 gether with bundle decompositions 푇 푀|푈 = 푇 푀 ⊕ 푇 푀 representing 풥 such that on the intersection 푈 ∩ 푉 of any 푈, 푉 ∈ 풰 we either have 푇 +,푈 푀 ⊂ 푇 +,푉 푀 and 푇 −,푈 푀 ⊃ 푇 −,푉 푀 or vice versa 푇 +,푈 푀 ⊃ 푇 +,푉 푀 and 푇 −,푈 푀 ⊂ 푇 −,푉 푀, and each inclusion is of finite codimension.

Proof. This follows directly from the proof of Proposition 2.53. Indeed, for each local decomposition over 푈 the Riesz projection is given by an integral over a circle of

+,푟푈 radius 푟푈 . For instance, if 푟푈 < 푟푉 and then Π is a projection onto a subspace of im Π+,푟푉 = 푇 +,푉 푀, thus 푇 +,푈 푀 ⊂ 푇 +,푉 푀. The finite codimension of this inclusion follows since Π+,푟푈 − Π+,푟푉 is the spectral projection onto the sum of generalized

eigenspaces 휆 between 푆−1(푟푈 ) and 푆−1(푟푉 ), which is finite-dimensional.

Finally, we can formulate decompositions in terms of “locally compatible” pointwise decompositions.

Corollary 2.55. Given 풥 there is a choice of pointwise decompositions 푇푚푀 = + − 푇푚 푀 ⊕ 푇푚 푀 which induce 풥푚 on each fiber 푇푚푀. Moreover, they can be chosen to be compatible in the sense that for every 푚 ∈ 푀 there exists an open neighborhood 푈 ⊂ 푀 with decomposition as in Corollary 2.54

A converse result is also true:

+ − Proposition 2.56. Suppose we are given decompositions 푇푚푀 = 푇푚 푀 ⊕ 푇푚 푀. Assume these are locally compatible, i.e., for every 푚 ∈ 푀 there exists an open

39 +,푈 −,푈 neighborhood 푈 ⊂ 푀 and a bundle decomposition 푇 푀|푈 = 푇 푀 ⊕ 푇 푀 with

+,푟푚 −,푟푚 the property that at each point 푥 ∈ 푈 the decompositions 푇푥푀 = 푇푥 푀 ⊕ 푇푥 푀 + − and 푇푚푀 = 푇푚 푀 ⊕ 푇푚 푀 induce the same polarization. Then there is a global polarization 풥 such that each restriction 풥푚 = 풥 |푇푚푀 is induced by the decomposition + − 푇푚푀 = 푇푚 푀 ⊕ 푇푚 푀.

Proof. Choose a locally finite open covering 풰 of 푀 and for each 푈 ∈ 풰 a decompo-

+,푈 −,푈 sition 푇 푀|푈 = 푇 푀 ⊕ 푇 푀 as in the hypothesis. For each such decomposition we get 퐽푈 ∈ 퐵(푇 푀|푈 , 푇 푀|푈 ) defined by 퐽푈 |푇 ±,푈 푀 = ±id. Take a partition of unity ∑︀ {휑푈 } subordinate to 풰. Then 퐽 = 푈∈풰 휑푈 퐽푈 represents the desired polarization.

Indeed, if we define 퐽 by 퐽 | ± = ±id. then 퐽 (푚) − 퐽 is compact for any 푚 푚 푇푚 푀 푈 푚

푚 ∈ 푈, so 휑푈 (푚)퐽푈 (푚) − 휑푈 (푚)퐽푚 is (well-defined and) compact for any 푚 ∈ 푀. ∑︀ It follows that 푈 휑푈 (푚)퐽푈 (푚) − 퐽푚 is compact for any 푚.

Applying the construction in the proof above verbatim to the local decompositions obtained in Corollary 2.54 we obtain the following result:

Corollary 2.57. For any polarization 풥 there exists a representative 퐽 ∈ 풥 such that at each point 푚 ∈ 푀 the map 퐽푚 : 푇푚푀 → 푇푚푀 is diagonalizable, has finite spectrum and all eigenvalues 휆 which are not ±1 have finite multiplicity.

Assume we have a polarization together with locally compatible decompositions

+ − 푇푚푀 = 푇푚 푀 ⊕ 푇푚 푀. This allows us to extend the definition of the index set (Definition 2.28) to 푀.

Definition 2.58 (index bundle). We define the index bundle of 푀 to be the set

⋃︁ ℐ(푀) = 퐼(푇푚푀) 푚∈푀 with the topology generated by the sets

{︁[︁ ]︁ 푉푠 = 푠|{푢}×푊 |푠 : 푈 × 푊 → 푇 푀|푈 , 푈 ⊂ 푀 open,

휋푀 푠(푢, 푤) = 푢, 푠 : {푢} × 푊 → 푇푢푀 is negative} ,

40 and the projection map 휋 : ℐ(푀) → 푀 mapping 퐼(푇푚푀) to 푚.

We first show that this gives a well-defined topology on ℐ(푀).

′ ′′ Lemma 2.59. For any 푠, 푠 as above, there exists 푠 such that 푉푠 ∩ 푉푠′ = 푉푠′′ .

′ ′ Proof. Let {푈푖} denote the connected components of 푈 ∩ 푈 and 푠푖 = 푠|푈푖 , 푠푖 = 푠|푈푖 . ⋃︀ We have 푉 ∩푉 ′ = 푉 ∩푉 ′ . We claim that for each 푖 either 푉 = 푉 ′ or 푉 ∩푉 ′ = . 푠 푠 푖 푠푖 푠푖 푠푖 푠푖 푠푖 푠푖 ∅ ′ Indeed, 휋(푉 ∩ 푉 ′ ) = {푢 ∈ 푈 : ind(푠 (푢 ), 푠 (푢 )) = 0} and by the continuity of the 푠푖 푠푖 푖 푖 푖 푖 푖 relative index and connectedness of 푈 we have that 휋(푉 ∩ 푉 ′ ) is either 푈 or , as 푖 푠푖 푠푖 푖 ∅ ′′ ′′ ᨀ wished. Finally we take 푠 to be 푠 restricted to 푈 = 푈푖. 푖:푉푠푖 =푉푠′ 푖

The index bundle ℐ(푀) is a principal bundle over Z – if we can show that it admits local trivializations.

Lemma 2.60. If the polarization of 푀 admits a trivialization over an open, connected set 푈, then there exists a trivialization ℐ(푀)|푈 ≃ 푈 ×Z which is Z-equivariant.

+,푈 −,푈 +,푈 −,푈 Proof. Let Φ: 푈 ×(푇푚 푀 ⊕푇푚 푀) → 푇 푀|푈 = 푇 푀 ⊕푇 푀 be a trivialization of the polarization. For any [푠] ∈ 퐼(퐻) we have a section 퐼푢,[푠] : 푢 ↦→ [Φ(푢, 푠)] which is continuous by Lemma 2.59. For the chosen decomposition ind−(Φ(푢, 푠)) = ind−(Φ(푚, 푠)) = ind− 푠 by continuity of Φ(푢, 푠(·)) in the 푢-variable and ind−(Φ(푢, 푠)+

1) = ind−(Φ(푢, 푠 + 1)) = ind−(푠 + 1) = ind− 푠 + 1 = ind−(Φ(푢, 푠)) + 1 as wished.

Finally, we aim to define indices globally over 푀 in order for any negative map to have a well-defined index.

Definition 2.61 (global index set). For a polarized manifold 푀, the set of global indices is defined as 퐼(푀) = 휋0(ℐ(푀)).

41 The group of integers Z acts transitively on 퐼(푀) and thus 퐼(푀) ≃ Z/(푑) for some 푑 ≥ 0. To determine this 푑 we consider the monodromy of the polarization.

Definition 2.62 (monodromy of a polarization). The monodromy

휑풥 : 휋1(푀) → Z of the principal Z-bundle ℐ(푀) is called the monodromy of 풥 .

Lemma 2.63. Let 퐽 ∈ 풥 represent the polarization. Then for a closed curve 훾 : 퐼 →

* * 푀, 훾(0) = 훾(1), the monodromy 휑풥 (훾) equals the spectral flow of 훾 퐽 : 훾 (푇 푀) → 훾*(푇 푀).

2 Proof. Without loss of generality we may assume 퐽 |훾(0) = 1 and thus we have + − * 푇훾(0)푀 = 푇훾(0)푀 ⊕ 푇훾(0)푀. Fix a trivialization 휑 : 훾 (푇 푀) ≃ 퐼 × 푇훾(0)(푀) as − * a bundle of polarized spaces. Let 푠 : 퐼 × 푇훾(0)(푀) → 훾 (푇 푀) be the map induced − * by the injection 퐼 × 푇훾(0)(푀) → 퐼 × 푇훾(0)(푀). We get a section [푠]: 퐼 → ℐ(훾 (푇 푀)) and the monodromy 휑풥 (훾) equals [푠(1)] − [푠(0)]. By Lemma 2.31 this equals the aforementioned spectral flow.

Lemma 2.64. Take 푑 ≥ 0 such that (푑) = im 휑풥 . Then 퐼(푀) ≃ Z/(푑) as a Z-torsor.

Proof. Let 푚0 be the basepoint of 푀 and assume 푘 ∈ im 휑풥 , that is 휑풥 (훾) = 푘 for some 훾. Then for any [푠] ∈ 퐼(푇푚0 푀) we have [푠] = [푠] + 푘 by the definition of the monodromy map.

On the other hand, if [푠] = [푠] + 푙 for some [푠] ∈ 퐼(푇푚0 푀) and 푙 ∈ Z, then by the definition of 퐼(푀) there is a curve 훾˜ : [0, 1] → ℐ(푀) such that 훾˜(0) = [푠], 훾˜(1) = [푠]+푙.

But then 휑풥 (휋 ∘ 훾˜) = 푙 and thus 푙 ∈ im 휑풥 .

Therefore for [푠1], [푠2] ∈ 퐼(푇푚0 푀) we get [푠1] = [푠2] in 퐼(푀) if and only if ind(푠1, 푠2) ∈ im 휑풥 , as wished.

42 Definition 2.65 (order of polarization; graded polarizations). We call 푑 the order of 풥 , or the order of 푀 as a polarized manifold, and denote by 푑풥 or 푑푀 . For

푑 dividing 푑푀 , a (Z/(푑)-)grading of 풥 is a Z-equivariant map 휑 : 퐼(푀) → Z/(푑). We call a pair (풥 , * ∈ 퐼(푀)) a graded polarization on 푀. (c.f. Definition 2.30).

Corollary 2.66. If 푀 is connected and 휋1(푀) = 0 then 퐼(푀) is a free Z-torsor, i.e., the action is free and transitive.

Having clearly established the relationship between abstract polarizations and concrete decompositions inducing them we move on to defining polarized chains and correspondences.

Definition 2.67 (polarized chains). Let (푀, 풥 ) be a polarized manifold and 푃 a manifold with corners. We say that 휎 : 푃 → 푀 is a polarized chain if 퐷휎푝 :

푇푝푃 → 푇휎(푝)푀 is a negative map for each 푝 ∈ 푃 (cf. Definition 2.24). In other + − words, at each 푝 ∈ 푃 for any compatible decomposition 푇휎(푝)푀 = 푇휎(푝)푀 ⊕ 푇휎(푝)푀 the map 휎 satisfies

− − + + Axiom N. Π 퐷휎푝 : 푇푝푃 → 푇휎(푝)푀 is Fredholm and Π 퐷휎푝 : 푇푝푃 → 푇휎(푝)푀 is compact,

We define the index map of 휎 to be 퐼(휎): 푃 → ℐ(푀), given by 퐼(휎)(푝) = [퐷휎푝]. Note that 휋 ∘ 퐼(휎) = 휎, i.e., 퐼(휎) lifts 휎 to the principal bundle ℐ(푀).

If 퐼(휎) is contained in the single component 퐼(푀) = 휋0(ℐ(푀)) then we say that ˜ 휎 is homogeneous and define the index via 퐼(휎) = [퐷휎푝] ∈ 퐼(푀). If 풥 = (풥 , 휑) is a (Z/(푑)-)graded polarization then we identify the index with the number ind(휎) = 휑(퐼(휎)) ∈ Z/(푑).

43 Definition 2.68 (polarized correspondence). Let (푀푖, 풥푖) be polarized mani- 푠 푡 folds, 푍 a manifold with corners and 푓 = (푓 , 푓 ): 푍 → 푀0 × 푀1 a smooth map. We say that 푓 is a polarized correspondence if for each 푧 ∈ 푍, 퐷푓푧 : 푇푧푍 →

푇푓 푠(푧)푀0⊕푇푓 푡(푧)푀1 is a linear correspondence (cf. Definition 2.35), i.e., at each 푧 ∈ 푍 + − and for each compatible pair of decompositions 푇푓 푠(푧)푀0 = 푇푓 푠(푧)푀0 ⊕ 푇푓 푠(푧)푀0 and + − 푇푓 푡(푧)푀1 = 푇푓 푡(푧)푀1 ⊕ 푇푓 푡(푧)푀1 one of the following equivalent axioms is satisfied:

+ 푠 − 푡 + − Axiom P1. For each 푧 ∈ 푍, the map (퐷 푓푧 , 퐷 푓푧): 푇푧푍 → 푇푓 푠(푧)푀0 ⊕ 푇푓 푡(푧)푀1 is + 푡 + 푠 + + 푠 Fredholm and the restriction 퐷 푓푧|ker 퐷 푓푧 : ker 퐷 푓푧 → 푇푓 푡(푧)푀1 is compact.

+ 푠 − 푡 + − Axiom P1’. For each 푧 ∈ 푍, the map (퐷 푓푧 , 퐷 푓푧): 푇푧푍 → 푇푓 푠(푧)푀0 ⊕ 푇푓 푡(푧)푀1 is + 푠 Fredholm. Moreover, if {푣푖} ⊂ 푇푧푍 is bounded and {퐷 푓푧 (푣푖)} is precompact, then + 푡 {퐷 푓푧(푣푖)} is precompact.

∨ We will denote it by 푓 : 푍 → 푀0 ×푀1. We say that 푓 is dense if (퐷푓)푧 : 푇푧푍 →

푇푓(푧)푀0 is dense at every point 푧 ∈ 푍. We define the index map of 푓 to be ℐ(푓)(푧) =

푠 푡 ℐ(퐷푓푧): 퐼(푇푓 (푧)푀0) × 퐼(푇푓 (푧)푀1) for any 푧 ∈ 푍. If 푍 is connected and 푑푀1 divides

푑푀0 , we define the global index of 푓 to be the induced map 퐼(푓): 퐼(푀0) → 퐼(푀1). ˜ If additionally 풥푖 = (풥푖, 휑푖) are Z/(푑)-graded polarizations then we define the index −1 ind 푓 = 휑2(퐼(푓)(휑1 (0))).

Therefore, whenever we refer to indices as taking integer values, we will always mean that the respective polarizations are graded.

Proposition 2.69 (polarized chains and polarized correspondences). A polar-

∨ ized chain 푓 : 푍 → 푀0 × 푀1 (of index 푘) is a polarized correspondence from 푀0 to

푀1 (of index 푘). A map 휎 : 푃 → 푀 is a polarized chain if and only if the corresponding map {0} × 휎 : {0} × 푃 → {0} × 푀 is a polarized correspondence.

44 Proof. It follows directly from the definitions of polarized chains and correspondences.

Proposition 2.70 (composition of polarized correspondences). Let 푓 : 푍 →

∨ ∨ 푀0 ×푀1 and 푔 : 푊 → 푀1 ×푀2 denote polarized correspondences for some polarized 푡 푠 manifolds 푀푖. Assume that 푓 is transverse to 푔 . Then the composition 푔 ∘ 푓 = ∨ 푓 ×푀1 푔 : 푍 ×푀1 푊 → 푀0 × 푀2 is a correspondence of polarized manifolds with index map ℐ(푔 ∘ 푓)(푧, 푤) = ℐ(푔)(푤) ∘ ℐ(푓)(푧), global index 퐼(푔 ∘ 푓) = 퐼(푔) ∘ 퐼(푓) and index ind(푔 ∘ 푓) = ind 푓 + ind 푔 whenever these make sense.

Proof. The fact that 퐷(푧,푤)(푔 ∘푓) is a linear correspondence at each (푧, 푤) ∈ 푍 ×푀1 푊 as well as the index formulas follow from Proposition 2.40 and Corollary 2.45.

Corollary 2.71 (composition of dense polarized correspondences). Let 푓 : 푍 →

∨ ∨ 푀0 × 푀1 and 푔 : 푊 → 푀1 × 푀2 denote polarized correspondences. Assume that 푔 is a dense polarized correspondence. Then 푓 푡 is transverse to 푔푠 and therefore 푔 ∘ 푓 is a correspondence of polarized manifolds. Moreover, if 푓 is a dense polarized correspondence, then 푔 ∘ 푓 is also a dense polarized correspondence.

Proof. As in the proof of Proposition 2.40, the operator (퐷+푓 푠, 퐷푓 푡 − 퐷푔푠, 퐷−푔푡) is Fredholm, and thus (퐷푓 푡 − 퐷푔푠) has a finite dimensional cokernel. We assumed that 퐷푔푠 is dense, so (퐷푓 푡 −퐷푔푠) is onto. Therefore 푓 푡 is transverse to 푔푠, and Proposition 2.70 implies 푔 ∘ 푓 is a correspondence. and gives the index formula. Proposition 2.40 implies that if 푓 is a dense correspondence then 푔 ∘ 푓 is also a dense correspondence.

2.5 Boundary of a manifold with corners

In order to define homology we will need a boundary operator on chains; thus,before introducing Floer chains, we shortly focus on rigorously defining boundary of manifolds with corners.

45 To define the boundary of a manifold with corners we need a refinement of the topological boundary operator 휕. For example, if one considers the simplex

푛 푛+1 Δ ⊂ R≥0 with the manifold with corners structure induced from the embedding, its topological boundary is, topologically, a sphere, but it is not an submanifold of

푛+1 R≥0 . We want to define the boundary in a way that will distinguish distinct faces of such a simplex, similarly to the boundary operator in singular homology. Let 푀 be an manifold with corners, and let 푚 ∈ 푆푘(푀). Each connected depth 푘

푘 chart 휑 = (푥1, . . . , 푥푘, 휓): 푈 → R≥0×퐻 around 푚 determines 푘 boundary components which we denote by (휑, 푖) for 푖 = 1, . . . , 푘, and which we will identify with the sets 푈 ∩

′ ′ ′ ′ {푥푖 = 0} for 푖 = 1, . . . , 푘. For any other connected depth 푘 chart 휑 = (푥1, . . . , 푥푘, 휓 ): ′ 푘 ′′ ′ 푈 → R≥0 × 퐻 we may choose a sufficiently small neighborhood 푈 ⊂ 푈 ∩ 푈 such ′ that both 휑|푈 ′′ and 휑 |푈 ′′ are connected charts, and therefore the transition map between them maps boundary components of 휑|푈 ′′ to the boundary components of ′ ′ 휑 |푈 ′′ , giving a bijection between boundary components of 휑 and 휑 . Thus, a choice of a boundary component for a connected chart around 푚 induces a compatible choice of a boundary component for any other connected chart around 푚.

Definition 2.72 (local boundary components). For each 푚 ∈ 푆푘(푀), a local boundary component of 푀 at 푚 is a compatible choice of a boundary component (휑, 푖) for each connected chart 휑 around 푚. We denote the local boundary component represented by a boundary component (휑, 푖) by [휑, 푖].

Notice that for each 푚 ∈ 푆푘(푀) there are exactly 푘 local boundary components at 푚.

Definition 2.73 (boundary of an manifold with corners). We define

휕푀¯ = {(푚, 훽)|훽 is a local boundary component at 푚}

46 and by 푖 : 휕푀¯ → 푀 we denote the local inclusion (푚, 훽) ↦→ 푚. 휑 ¯ Each boundary component (휑, 푖) induces a set 휕푖 푈 ⊂ 휕푀 given by pairs ′ ′ 휑 (푚 , [휑, 푖]) for every 푚 ∈ 푈 ∩ {푥푖 = 0}. Thus, there is also a bijection 휕푖 푈 → ′ ′ ¯ 푈 ∩ {푥푖 = 0} given by (푚 , [휑, 푖]) ↦→ 푚 . The topology of 휕푀 is given by requiring that these maps be homeomorphisms onto image. The stratification on 휕푀¯ is induced from 푀, i.e., 푆푘(휕푀¯ ) = 푖−1(푆푘+1(푀)).

휑 푘−1 Finally, the maps 휕푖휑 = (푥1,..., 푥^푖, . . . , 푥푘, 휓): 휕푖 푈 → R≥0 × 퐻 are taken as charts on 휕푀¯ .

2.6 Floer spaces and correspondences

We combine the notions of semicompact and polarized spaces, maps and correspondences to define the Floer spaces, chains and correspondences. The chains willbe later used to define homology of Floer spaces and correspondences – to define maps on homology.

Definition 2.74 (Floer space). A Floer space is a triple X = (푋, ℒ, 풥 ) where 푋 is a Hilbert manifold, ℒ is a Floer functional and 풥 is a polarization of 푋. The dual Floer space is the triple X∨ = (푋∨, ℒ∨, 풥 ∨) (i.e., 푋∨ = 푋, ℒ∨ = −ℒ and 풥 ∨ = −풥 ).

Example 2.75. The trivial Floer space is O = ({0}, ℒO = 0, 풥O = {0}).

Definition 2.76 (product of Floer spaces). If X, Y are Floer spaces, we define

the floer space X × Y to be the triple (푋 × 푌, ℒ푋×푌 , 풥푋×푌 ), where ℒ푋×푌 (푥, 푦) =

ℒ푋 (푥) + ℒ푌 (푦) and 풥푋×푌 = 풥푋 ⊕ 풥푌 .

47 Definition 2.77 (Floer chain). A Floer chain (or a chain, for short) in X is a manifold with corners 푃 and a smooth map 휎 : 푃 → 푋 which a semi-compact map with respect to ℒ (cf. Definition 2.5) and a polarized chain (cf. Definition 2.67). We denote it by 휎 : 푃 → X.

− We say that the chain 휎 has index 푘 if 퐷푝 휎 has index 푘 at each point of the interior of 푃 . We denote it by ind 휎 = 푘. The boundary 휕휎 is defined to be the manifold with corners 휕푃¯ together with the map 휕휎 = 휎 ∘ 푖 where 푖 : 휕푀¯ → 푀 is the local inclusion map (cf. Definition 2.73).

Recall that being a semi-compact map and a polarized chain means satisfying Axioms (M1, M2, M3, N):

Axiom M1. On the weak closure of 휎(푃 ), the function ℒ is bounded above and upper semi-continuous for the weak topology.

Axiom M2. Any subset 푆 ⊂ im 휎 on which ℒ is bounded is precompact for the weak topology.

Axiom M3. If 휎(푥푖) ⇀ 푦 and lim ℒ(휎(푥푖)) = ℒ(푦), then 푥푖 has a (strongly) convergent subsequence.

− − + + Axiom N. Π 퐷휎푝 : 푇푝푃 → 푇휎(푝)푀 is Fredholm and Π 퐷휎푝 : 푇푝푃 → 푇휎(푝)푀 is compact,

Definition 2.78 (Floer correspondence). A (Floer) correspondence from X1 to 푠 푡 X2 is an manifold with corners 푍 and a smooth map 푓 = (푓 , 푓 ): 푍 → 푋1 ×푋2 such

that 푓 is a semi-compact correspondence from (푋1, ℒ1) to (푋2, ℒ2) (cf. Definition

2.6) and a polarized correspondence from from (푋1, 풥1) to (푋2, 풥2) (cf. Definition ∨ 2.68). We denote it by 푓 : 푍 → X1 × X2.

48 A dense correspondence is a correspondence which is dense as a polarized correspondence, i.e. a correspondence for which 퐷푓 푠 is dense at every point.

The index map ℐ(푓): 푍 → HomZ퐼(푇푓 푠(푧)푋1) → 퐼(푇푓 푡(푧)푋2) and global index

퐼(푓): 퐼(X1) → 퐼(X2) of a Floer correspondence 푓 are defined as for the coresponding polarized correspondence 푓.

Recall that being a semi-compact correspondence and a polarized correspondence means satisfying Axioms (C1), (C2), (C3), and either Axiom (P1) or, equivalently, Axiom (P1’):

Axiom C1. On the weak closure of im 푓, the function ℒ = ℒ2 − ℒ1 is bounded above and upper semi-continuous for the weak topology.

푠 푡 Axiom C2. If 푓 (푧푖) is weakly precompact and ℒ2(푓 (푧푖)) is bounded below, then 푡 푓 (푧푖) is weakly precompact.

푠 푡 Axiom C3. If 푓 (푧푖) → 푥 and 푓 (푧푖) ⇀ 푦 and ℒ(푓(푧푖)) → ℒ(푥, 푦), then 푧푖 has a (strongly) convergent subsequence.

Remark 2.79. Note that 휎 : 푃 → 푋 is a chain in X if and only if it is a correspondence from the trivial Floer space O to X.

Correspondences may arise as chains in product Floer spaces.

∨ Lemma 2.80. Suppose 휎 : 푃 → 푋1 × 푋2 is a chain in X1 × X2 (of index 푘 ∈ Z). ∨ Then 휎 is a correspondence from X1 to X2 (of index 푘), and 휎 : 푃 → 푋2 × 푋1 given ∨ ∨ by swapping the coordinates of 휎 is a correspondence from X2 to X1 (of index 푘).

49 In this case the correspondence obtained also gives a “dual correspondence” going

∨ ∨ from X2 to X1 .

푠 푡 Definition 2.81 (dual map). Let 퐹 = (퐹 , 퐹 ): 푍 → 푋1 × 푋2 be a smooth map. We denote by 퐹 ∨ the dual or transpose map 퐹 ∨ = (퐹 ∨푠, 퐹 ∨푡) = (퐹 푡, 퐹 푠): 푍 →

푋2 × 푋1.

Example 2.82. The diagonal map Δ: 푋 → 푋∨ × 푋 is a dense correspondence but it is not a chain in X∨ × X unless 푋 is compact.

Definition 2.83 (composition of Floer correspondences). Given a corre-

∨ ∨ spondence 푓 : 푍 → X0 × X1 and a correspondence 푔 : 푊 → X1 × X2 transverse to 푡 푠 ∨ it (i.e., 푓 t 푔 ), we define the composition 푔 ∘ 푓 = 푓 ×B1 푔 : 푍 ×X1 푊 → X0 × X2, given by (푤, 푧) ↦→ (푓 푠(푧), 푔푡(푤)).

Proposition 2.84. The composition 푔 ∘ 푓 = 푓 ×B1 푔 of a correspondence 푓 : 푍 → ∨ ∨ X0 × X1 and a correspondence 푔 : 푊 → X1 × X2, transverse to it is a correspondence of Floer spaces, with index map ℐ(푔 ∘ 푓)(푧, 푤) = ℐ(푔)(푤) ∘ ℐ(푓)(푧), global index 퐼(푔 ∘ 푓) = 퐼(푔) ∘ 퐼(푓) and index ind(푔 ∘ 푓) = ind 푓 + ind 푔 whenever these are defined. Moreover, if 푓 is dense, then 푔 ∘ 푓 is also dense.

Proof. This composition is a semi-compact correspondence due to Proposition 2.8, and a polarized correspondence due to Proposition 2.70.

The denseness statement follows from Corollary 2.71.

Corollary 2.85. If 휎 : 푃 → X1 is a chain transverse to a correspondence 푓 : 푍 → ∨ 푠 X1 × X2 (i.e., 휎 t 푓 ), then the composition 푓 ∘ 휎 : 푃 ×푋1 푍 → X2 is a chain with index map ℐ(푓 ∘ 휎)(푝, 푧) = ℐ(푓)(푧) ∘ ℐ(휎)(푝) and index 퐼(푓 ∘ 휎) = 퐼(푓) ∘ 퐼(휎) and ind(푓 ∘ 휎) = ind 휎 + ind 푓.

50 In some cases a dense correspondence will be actually a graph of a (possibly parametrized) map between two Floer spaces.

Proposition 2.86. Let 퐹 : 푋 × 푃 → 푌 with 푃 finite dimensional and compact be

a smooth, weakly continuous map. Then its parametrized graph Γ퐹,푃 (퐹 ): 푋 × 푃 →

푋 × 푌 defined by Γ퐹,푃 (푥, 푝) = (푥, 퐹 (푥, 푝)) is a Floer correspondence from X to Y if

and only if ℒ푌 ∘ 퐹 − ℒ푋 : 푋 → R is bounded above and weakly upper semi-continuous − and for each (푥, 푝) ∈ 푋 ×푃 the operator 퐷퐹 | − : 푇 푋 → 푇퐹 (푥,푝)푌 is a negative 푇(푥,푝)푋 (푥,푝) map.

Proof. We restrict ourselves to the unparametrized case for simplicity of exposition. Since 푃 is assumed to be finite dimensional and compact, the same proof goes through with minor adjustments in the 푃 -parametrized case. Since 퐹 is continuous and weakly continuous, Axioms (C2) and (C3) are auto-

matically satisfied for Γ퐹 .

Since 퐹 is weakly continuous, ℒ푌 − ℒ푋 is bounded above on the weak closure of

im Γ퐹 if and only if ℒ푌 ∘ 퐹 − ℒ푋 is bounded above on 푋. Similarly, if (푥푖, 퐹 (푥푖)) ⇀

(푥, 푦) then 퐹 (푥푖) ⇀ 퐹 (푥) = 푦 and therefore lim sup(ℒ푌 −ℒ푋 )(푥푖, 퐹 (푥푖)) ≤ (ℒ푌 − ℒ푋 )(푥, 푦) if and only if lim sup(ℒ푌 ∘ 퐹 − ℒ푋 )(푥푖) ≤ (ℒ푌 ∘ 퐹 − ℒ푋 )(푥). 푠 Finally, regarding Axiom (P1), we notice that 퐷Γ퐹 (푣) = (푣, 퐷퐹 (푣)), with Γ퐹 = 푡 + 푠 − 푡 + − + − id푋 and Γ퐹 = 퐹 . Therefore 퐷 Γ퐹 ⊕ 퐷 Γ퐹 : 푇 푋 ⊕ 푇 푋 → 푇 푋 ⊕ 푇 푌 is equal ⎛ ⎞ id푇 +푋 0 to ⎜ ⎟ and Proposition 2.43 implies that this is Fredholm if and ⎝ − − ⎠ 퐷 퐹 |푇 +푋 퐷 퐹 |푇 −푋 − + 푠 + − only if 퐷 퐹 |푇 −푋 is Fredholm. On the other hand, ker 퐷 Γ퐹 = ker 퐷 id푇 푋 = 푇 푋 + 푡 + and thus 퐷 Γ | + 푠 = 퐷 퐹 | − . We conclude that Axiom (P1) is satisfied for 퐹 ker(퐷 Γ퐹 ) 푇 푋

Γ if and only if 퐷퐹 | − is a negative map for every 푥 ∈ 푋. 퐹 푥 푇푥 푋

+ 푠 − 푡 The characterization of 퐷 Γ퐹 ⊕ 퐷 Γ퐹 given in the proof above also allows one to compute the ind Γ퐹 directly:

∨ Lemma 2.87. If Γ퐹,푃 is a Floer correspondence from X to Y (resp. Γ퐹,푃 is a Floer − ∨ + correspondence from Y to X) then ind Γ퐹 = ind 퐷 퐹 |푇 −푋 (resp. ind Γ퐹 = ind 퐷 퐹 |푇 +푋 ).

51 52 Chapter 3

Homology of Floer Spaces

This chapter introduces the homology of Floer spaces and maps induced by correspondences, as well as the Poincaré pairing between the homology of a Floer space and that of its dual. Immediately after that we compute the homology groups in the case of finite-dimensional manifolds. Dense correspondences always induce maps on homology, but fot the ones that are not dense as well as to define the Poincaré pairing we need to be able toperturb chains to be transverse to correspondences or to chains of the dual Floer space. We introduce a set of axioms that are sufficient for a family of perturbations to achieve these goals. With perturbations with hand we proceed to proving suspension isomorphisms. Then we finish the chapter by defining a class of perturbations of the Floer functional that preserves the chains and thus preserves semiinfinite homology.

3.1 Defining homology

In this section we define homology of Floer spaces over the field of F2. We assume that the Floer space 푋 is simply-connected throughout for simplicity of exposition.

Definition 3.1 (negligibility). A set 푆 ⊂ 푋 is 푘-negligible if it is contained in the image of a polarized chain 휂 : 푄 → 푋 of index ind 휂 < 푘.

53 A chain 휎 : 푃 → 푋 of index 푘 is negligible if im 휎 is 푘-negligible.

Definition 3.2 (homology of Floer spaces). Let 퐶퐹̃︂ 푘(X) be the F2-vector space generated by isomorphism classes of chains of index 푘 modulo the relations 휎 + 휂 = 휎 ⊔ 휂.

Define 휕 : 퐶퐹̃︂ 푘(X) → 퐶퐹̃︂ 푘(X) by 휕[휎] = [휎|휕푃¯ ] for 휎 : 푃 → 푋.

Let 푁푘(X) ⊂ 퐶퐹̃︂ 푘(X) be the subgroup generated by negligible chains 휎 : 푃 →

푋 such that 휕휎 is also negligible (i.e. (푘 − 1)-negligible), and define 퐶퐹푘(X) =

퐶퐹̃︂ 푘(X)/푁푘(X).

Define 퐻퐹푘(X) = 퐻푘(퐶퐹*(X), 휕) to be the homology of this complex.

Note that the relation 휎 + 휂 = 휎 ⊔ 휂 implies that every element in 퐶퐹푘(X) can be represented as a (single) chain. To prove that the definition of the homology groups is correct, one needs tocheck that 휕 descends to 퐶퐹푘(X). But 휕(푁푘(X)) ⊂ 푁푘−1(X) follows easily since we assumed 2 the boundary of a chain in 푁푘(X) is negligible as well and 휕 = 0.

Definition 3.3 (relative homology of Floer spaces). Let 퐴 ⊂ 푋 be any subset. We define 퐶퐹푘(퐴) = {휎 ∈ 퐶퐹푘(X) : im 휎 ⊂ 퐴}. Since 퐶퐹*(퐴) is a subcomplex of 퐶퐹*(X), we can define

퐻퐹푘(퐴) = 퐻푘(퐶퐹*(퐴), 휕) as well as

퐻퐹푘(X, 퐴) = 퐻푘(퐶퐹*(X)/퐶퐹*(퐴), 휕).

In Section 3.3 we prove Theorem 4 from which it follows that a correspondence, given a sufficient family of perturbations, induces a map on homology:

54 Corollary 3.4. Let 퐺 be a correspondence from X to Y. If there exists a 퐺-adapted family of perturbations, then 퐺 induces a map 퐻퐹*(퐺): 퐻퐹*(X) → 퐻퐹*+ind 퐺(Y) which on a cycle 휎 transverse to 퐺 is given by taking the fiber product 휎 ×푋 퐺.

In particular,

Corollary 3.5. A dense correspondence 퐺 from X to Y induces a map 퐻퐹*(퐺):

퐻퐹*(X) → 퐻퐹*+ind 퐺(Y) which on a cycle 휎 is given by taking the fiber product

휎 ×푋 퐺.

3.2 Homology in finite dimensions

We proceed to analyze a simple class of examples of Floer spaces, i.e., when the Floer spaces at hand are actually finite-dimensional. If the functional satisfies the Palais-Smale condition then the resulting homology is isomorphic to Morse homology.

Proposition 3.6. Assume that 푋 is finite-dimensional. Then 휎 : 푃 → X is a chain if and only if:

• 푃 is finite dimensional,

• ℒ ∘ 휎 is bounded above,

• ℒ ∘ 휎 is proper.

Proof. Since in this case the weak topology on 푋 coincides with the strong topology, it is straightforward to check that Axiom (M1) is equivalent to ℒ ∘ 휎 being bounded above while Axioms (M2) and (M3) are equivalent to ℒ ∘ 휎 being proper. Moreover, since 푇 푋 is finite dimensional, 퐷휎 is a negative map if and only if 푇 푃 is finite dimensional.

Notice, in particular, that the choice of the polarization for a finite dimensional manifold does not change the geometric chain complex we obtain besides possibly introducing a shift in grading. The natural thing to do is to always take 푇 −푋 = {0}

55 and 푇 +푋 = 푇 푋 so that chains in grading 푘 are exactly the chains 휎 : 푃 → 푋 for which 푃 is finite-dimensional, just as it is done in singular homology and some geometric homology theories. The situation simplifies further in the compact case.

Corollary 3.7. If 푋 is compact, then 휎 : 푃 → 푋 is a chain if and only if 푃 is finite dimensional and compact.

푔푚 In particular, 퐶퐹*(푋, ℒ) does not depend on ℒ and is actually the same as 퐶* (푋) (assuming 푇 −푋 = {0}), the chain complex of geometric chains as defined by Lipyan- skiy, [Lip08,Lip14].

If the functional is bounded below, the chain complex also depends only on the topology of 푋.

Proposition 3.8. Let 푋 be finite dimensional, ℒ be bounded below and take 푇 −푋 =

{0}. Then there is a natural isomorphism 퐻퐹푘(X) ≃ 퐻푘(푋).

Proof. For any chain 휎 ∈ 퐶퐹푘(X), we have that ℒ ∘ 휎 is bounded both above and below, and by properness of ℒ ∘ 휎 we get that 푃 is compact. But for any compact 푃 the map 휎 : 푃 → X is a chain, so we get that Floer chains are exactly the geometric

푔푚 chains, 퐶퐹*(X) = 퐶* (푋). Since geometric homology is naturally isomorphic to ordinary homology ([Lip14]), the result follows.

− Proposition 3.9. Let 푋 be finite dimensional and 푇 푋 = {0}. Then 퐻퐹푘(X) ≃ lim 퐻 (푋, ℒ−1((−∞, 퐶])) where the limit is taken over maps 휄 : 퐻 (푋, ℒ−1((−∞, −퐶])) → ←− 푘 퐶퐷 푘 −1 퐻푘(푋, ℒ ((−∞, −퐷])) for all 퐶 ≥ 퐷 ≥ 0.

Proof. Let 0 ≥ −퐶푖 → −∞ be a decreasing sequence of regular values of ℒ with lim 퐶푖 = ∞ and denote 휄푖 = 휄퐶푖퐶푖−1 −1 We start by defining maps Φ푖 : 퐻퐹푘(X) → 퐻푘(푋, ℒ ((−∞, −퐶푖]). compatible with all 휄푖. Take a class [휎] ∈ 퐻퐹푘(X). One can perturb 휎 to be transverse to all −1 −1 ℒ (−퐶푖) and represent the same homology class. Taking 푃푖 = (ℒ ∘ 휎) ([−퐶푖, ∞)) −1 we get that 휎|푃푖 is a cycle in 퐻푘(푋, ℒ ((−∞, −퐶푖]) and define Φ푖([휎]) = [휎|푃푖 ].

56 Using this restriction argument one also shows that Φ푖 is well-defined. Moreover 휎| is homologous to 휎| ∪ 휎| but 휎| is zero in 퐻 (푋, ℒ−1((−∞, −퐶 ]). 푃푖 푃푖−1 푃푖∖푃˚푖 푃푖∖푃˚푖 푘 푖−1 Therefore 휄 Φ = Φ and we obtain a map Φ: 퐻퐹 (X) → lim 퐻 (푋, ℒ−1((−∞, 퐶]). 푖 푖 푖−1 푘 ←− 푘 We proceed to constructing the inverse of this map. Suppose we are given a se-

′ ′ quence 휎1, 휎2,... representing an element in the limit. Without losing generality we ′ −1 ′ may assume 휎푖 is transverse to ℒ (−퐶푗) for each 푗. Let us take 휎1 = 휎1 and construct ′ ′ a sequence {휎푖} inductively as follows. Take 휎푖−1, 휎푖 such that 휄푖([휎푖]) = [휎푖−1]. By as- −1 ′ sumption, there is a chain 휏 in (푋, ℒ ((−∞, −퐶푖−1]) such that 휕휏 = 휎푖−1 − 휎푖 + 훿푖−1 푔푚 −1 with 훿푖−1 ∈ 퐶푘 (ℒ ((−∞, −퐶푖−1])), and without losing generality we may assume −1 2 휏 is transverse to ℒ (−퐶푗) for each 푗. Take 휎푖 to be 휎푖−1 ∪ 훿푖−1. Since 휕 휏 = 0, ′ −1 ′ we get that 휕휎푖 = 휕휎푖 and therefore [휎푖] ∈ 퐻푘(푋, ℒ ((−∞, −퐶푖]) with [휎푖] = [휎푖].

Define 휎 = 휎1 ∪ 훿1 ∪ 훿2 ∪ .... Since 휎1 and each 훿푖 are compact and ℒ ∘ 훿푖 ≤ −퐶푖 we get that 휎 is a chain in X. Moreover, by construction we get that ℒ|휕휎 ≤ −퐶푖 ′ for each 푖 and therefore 휕휎 = 0. We define Ψ({[휎푖]}) = [휎] and this gives a map Ψ : lim 퐻 (푋, ℒ−1((−∞, 퐶]) → 퐻퐹 (X). ←− 푘 푘 By construction we get that Φ ∘ Ψ is an identity. The composition Ψ ∘ Φ is

−1 an identity because each cycle 휎 in X which is transverse to all ℒ (−퐶푖) can be

−1 −1 subdivided into 휎1 = 휎|ℒ ([−퐶1,∞)) and 훿푖−1 = 휎|ℒ ([−퐶푖,퐶푖−1]).

If the functional is behaving well, the topology of the finite-dimensional Floer space is determined by a suitable compactification. Let 푋−,ℒ = 푋 ∪ {−∞} with the topology given by the open sets of 푋 together with sets ℒ−1((−∞, 퐶)) ∪ {−∞}. One such instance is when the end of the Floer space can be made cylindrical.

Corollary 3.10. Let 푋 be finite dimensional and 푇 −푋 = {0}. Assume that for some 퐶 there is a homeomorphism 퐹 : ℒ−1((−∞, −퐶]) → (−∞, −퐶] × 푌 such that the

−,ℒ projection to the first coordinate is equal to ℒ. Then 퐻퐹푘(X) ≃ 퐻푘(푋 , {−∞}).

Proof. The assumptions imply that there is a neighborhood of {−∞} which is the cone on ℒ−1(−퐶), with {−∞} being the tip of the cone. Therefore the projection 푋−,ℒ → 푋−,ℒ/ ({−∞} ∪ ℒ−1((−∞, −퐷])) for 퐷 ≥ 퐶 is a homotopy equivalence, and

−1 this last space is homeomorphic to 푋/ℒ ((−∞, −퐷]).

57 Corollary 3.11. Let 푋 be a finite dimensional, complete Riemannian manifold, with

− 푇 푋 = {0}. Let Φ푡 denote the flow of −∇ℒ on 푋. Assume that there exists 퐶 −1 such that for every 푥 ∈ ℒ ((−∞, −퐶]) we have lim푡→∞ ℒ(Φ푡(푥)) = −∞. Then −,ℒ 퐻퐹푘(X) ≃ 퐻푘(푋 , {−∞}).

Proof. In such a case one can use the flow of the normalized gradient to explicitly construct the diffeomorphism which is required to apply Corollary 3.10.

This covers the case of a Palais-Smale function which has critical set bounded below.

Corollary 3.12. Let 푋 be a finite dimensional, complete Riemannian manifold, with 푇 −푋 = {0}. Assume that ℒ is bounded below on the set of critical points of ℒ and that

ℒ satisfies the Palais-Smale condition, i.e., any sequence {푥푖} with {ℒ(푥푖)} bounded −,ℒ and lim푖→∞ ∇ℒ(푥푖) = 0 is precompact. Then 퐻퐹푘(X) ≃ 퐻푘(푋 , {−∞}).

Proof. Since we assumed ℒ is bounded below on the set of critical points of ℒ, therefore there is 퐶 such that there are no critical points in ℒ−1((−∞, 퐶]).

−1 Suppose there is 푥 ∈ ℒ ((−∞, 퐶]) such that lim푡→∞ ℒ(Φ푡(푥)) = 퐷 > −∞. Then

{ℒ(Φ푡(푥))}푡≥0 is bounded and lim푡→∞ ∇ℒ|Φ푡(푥) = 0, so by the assumptions {Φ푡(푥)}푡≥0 is precompact. But if lim푡푖→∞ Φ푡푖 (푥) = 푥0, then ∇ℒ(푥0) = 0 and ℒ(푥0) < 퐶, which contradicts the face that there are no critical points in ℒ−1((−∞, 퐶]).

Now the result follows from the previous Corollary, 3.11.

One can also ask about what happens for a Palais-Smale function which may have an unbounded critical set. Proposition 3.9 can be then applied to show that we do recover the Morse homology.

Corollary 3.13 (Palais-Smale function in finite dimensions). Let 푋 be a finite dimensional, complete Riemannian manifold, with 푇 −푋 = {0}. Assume ℒ satisfies the Palais-Smale condition, i.e., any sequence {푥푖} with {ℒ(푥푖)} bounded and lim푖→∞ ∇ℒ(푥푖) = 0 is precompact. Also assume that ℒ is Morse-Smale, so that −,ℒ Morse homology of 푋 is well-defined. Then 퐻퐹푘(X) ≃ 퐻푘(푋 , {−∞}).

58 Proof. The Palais-Smale condition allows us to choose a decreasing sequence 퐶푖 → −1 −∞ such that ℒ (퐶푖) do not contain any critical points of ℒ. Therefore for each 푖 푀 −1 we have that the Morse homology 퐻* (푋 ∖ ℒ ((−∞, 퐶푖))) is isomorphic to the ordi- −1 nary homology 퐻*(푋 ∖ ℒ ((−∞, 퐶푖))). Moreover, these isomorphism commute with 푀 −1 the maps on homology induced by the quotient maps 퐶* (푋 ∖ ℒ ((−∞, 퐶푖+1))) → 푀 −1 퐶* (푋 ∖ ℒ ((−∞, 퐶푖))) and with maps 휄퐶푖+1퐶푖 on the side of ordinary homology. Therefore 퐻퐹 (X) ≃ lim 퐻 (푋, ℒ−1((−∞, 퐶])) ≃ lim 퐻푀 (푋, ℒ−1((−∞, 퐶])) 푘 ←− 푘 ←− 푘 ≃ 퐻 (lim 퐶푀 (푋, ℒ−1((−∞, 퐶])) ≃ 퐻 (퐶푀 (푋)) ≃ 퐻푀 (푋) since by the Morse-Smale 푘 ←− * 푘 * 푘 condition there is only a finite number of trajectories coming out of a chosen critical point counted by the Morse differential.

We now turn to some explicit computations, starting with the computation of the homology of a single point (with the unique polarization and any functional).

Proposition 3.14. 퐻퐹0({푝}) = F2 and 퐻퐹푘({푝}) = 0 for 푘 ̸= 0.

Proof. The result is obvious for 푘 ≤ 0.

For 푘 ≥ 2, every chain of index 푘 is negligible, so 퐶퐹푘({푝}) = 0.

For 푘 = 1, not every chain is negligible, but every cycle in 퐶퐹1({푝}) is negligible.

Thus 퐻퐹1({푝}) = 0.

The simplest noncompact space we may encounter is the real line R. We compute its homology for an arbitrary functional ℒ and the trivial polarization 푇 −R = {0}.

− Proposition 3.15. Let 푋 = R and 푇 푋 = {0}. Let us denote 푟± = lim sup푡→±∞ ℒ(푡) ∈ [−∞, +∞].

1. If 푟− > −∞, 푟+ > −∞ and ℒ is bounded below, then 퐻퐹푘(R, ℒ) = 퐻퐹푘({0}).

2. If 푟∓ = −∞ and 푟± > −∞, and ℒ|[0,±∞) is bounded below, then 퐻퐹푘(R, ℒ) = 0.

3. If 푟− = 푟+ = −∞, then 퐻퐹푘(R, ℒ) = 퐻퐹푘−1({0}).

Remark 3.16. The results above hold without the assumptions of ℒ being bounded below, but this assumption simplifies the exposition greatly.

59 Proof. 1. In such case each chain 휎 : 푃 → R is necessarily compact, and vice versa, every smooth map from a compact manifold with corners is a chain.

Therefore 퐻(휎): 푃 × [0, 1] → R defined as 퐻(휎)(푝, 푡) = 푡 · 휎(푝) is also a chain and 휕퐻(휎) = −휎 + 0 · 휎 + 퐻(휕휎). Therefore 퐻 is a chain homotopy between

identity on 퐶퐹*(R, ℒ) and the map 휎 ↦→ 0 · 휎. What follows is that 퐶퐹*(R, ℒ)

is chain homotopic to its subcomplex 퐶퐹*({0}).

2. Without loss of generality, assume 푟+ = −∞ and 푟− > −∞. This means that a smooth map 휎 is a chain if and only if it is proper and the set 휎−1((−∞, 0])

is compact. What follows is that 퐻(휎): 푃 × [0, ∞) → R given by 퐻(휎)(푝, 푡) = 휎(푝) + 푡 is a chain. If 휎 is a cycle we get 휕퐻(휎) = 휎 which finishes the proof of this result.

3. In this case ℒ is bounded above and proper, therefore a smooth map 휎 is

2 a chain if and only if it is proper. So we get that 퐶퐹*(R, ℒ) = 퐶퐹*(R, −푥 ). We compute the homology for this Floer space in the upcoming Proposition

3.22, modulo a shift by 1 in degree coming from a change of polarization.

3.3 Perturbing chains

In this section we define admissible perturbations and show that their existence implies possibility of suitably perturbing chains to achieve transversality. Moreover, we define adapted perturbations which are used to show suspension isomorphisms for products X × 퐻 without assuming anything about perturbations in X. In any case, the main point of concern in this section is to make sure that after perturbing a chain one still obtains a chain. We want to be able to achieve transversality between arbitrary chains in X and arbitrary chains in X∨ in order to be able to define a pairing between the corresponding semi-infinite homologies. Also, given a correspondence from X to Y which is not dense, we may want to achieve transversality between arbitrary chains in X and this correspondence in order to define an induced map between semi-infinite homologies of

60 X and Y. For compact finite dimensional manifolds this can easily be done using ar- bitrarily chosen perturbations, but in our case the Floer space is neither bounded nor finite-dimensional and therefore we need to be somewhat careful when constructing the perturbations. In this section we will define an abstract notion of a family of admissible perturbations. We need to construct specific families for each specific case, and later onwe will construct such a family for Seiberg-Witten-Floer spaces.

Definition 3.17 (admissible perturbations). Let P be the open unit ball in a Hilbert space. A smooth map Θ: 푋 × P → 푋 is called a family of admissible

perturbations for X if Θ|푋×{0} = id푋 and it satisfies the following axioms:

Axiom P1. Suppose 푝푖 → 푝. Then 푥푖 ⇀ 푥 for some 푥 if and only if Θ(푥푖, 푝푖) ⇀ 푥˜ for some 푥˜. Moreover, in this case 푥˜ = Θ(푥, 푝).

Axiom P2. There exists a smooth function 푙 : R≥0 → R≥1 such that |ℒ(Θ(푥, 푝)) − ℒ(푥)| ≤ ‖푝‖· 푙(‖푥‖2).

Axiom P3. If 푥푖 ⇀ 푥 in 푋 and 푝푖 → 푝 ⊂ P, then

ℒ(Θ(푥푖, 푝푖)) − ℒ(푥푖) → ℒ(Θ(푥, 푝)) − ℒ(푥).

Axiom P4. 퐷푋 Θ(푥,푝) ∈ ℱ푟푒푠(푇푥푋, 푇Θ(푥,푝)푋) for each (푥, 푝) ∈ 푋 × P.

Axiom P5. 퐷PΘ(푥,푝) is compact for each (푥, 푝) ∈ 푋 × P.

Axiom P6. 퐷PΘ(푥,푝) has dense image in 푇푥푋 for each (푥, 푝) ∈ 푋 × P.

If we want to achieve transversality with a given correspondence 퐹 we do not need such a strong family of perturbations. We only need to perturb along the image of 퐹 푠 and only in the directions complementary to 퐹 푠.

61 Definition 3.18퐹 ( -adapted perturbations). Let 퐹 : 푍 → 푋∨ × 푌 be a correspondence from X to Y). A smooth map Θ: 푋 × P → 푋 is called a (퐹 -adapted) family of admissible perturbations if it satisfies Axioms (P1, P2, P3, P4, P5) and the following axiom instead of Axiom (P6):

Axiom P6.˜ For each 푥 ∈ 푋 the map Θ(푥, ·) is transverse to 퐹 푠.

We now show that existence of admissible perturbations as defined above gives possibility of perturbing chains in arbitrary directions guaranteeing that the perturbed maps are also chains.

Theorem 1. Let Θ be a family of admissible perturbations for X. Suppose 휎 : 푃 → X is a chain and 퐴 ⊂ 푃 is a closed subset, and there exists a smooth

function 휂 : 푃 → R such that 휂|퐴 ≡ 0 and for any 푦 ∈ 푃 ∖퐴 we have 휂(푝) ∈ (0, 1]. (In particular, for 퐴 = ∅ one can take 휂 ≡ 1.) Then there is a smooth map

퐹 = 퐹휎,Θ : 푃 × P → 푋 such that:

1. 퐹 extends 휎 over 푃 × {0} ∪ 퐴 × P, i.e., for any (푦, 푝) ∈ 푃 × {0} ∪ 퐴 × P we have 퐹 (푦, 푝) = 휎(푦).

2. For each (푦, 푝) ∈ (푃 ∖ 퐴) × P the differential 퐷P퐹(푦,푝) has dense image.

3. At each point (푦, 푝) ∈ 푃 × P the differential 퐷푃 퐹(푦,푝) is a negative map and

퐷P퐹(푦,푝) is compact.

4. If 퐾 ⊂ P is a compact manifold with corners (in particular, finite dimen-

sional), then 휎퐾 = 퐹 |퐾×푃 is a chain in X.

휂(푦) Proof. We define 훾(푦) = 푙(‖푦‖2) and 퐹휎,Θ(푦, 푝) = Θ (휎(푦), 훾(푦)푝), and claim that 퐹 satisfies the properties listed in the Theorem.

62 The first property follows since for any 푥 ∈ 푋, Θ(푥, 0) = 푥.

The second property follows from Axiom (P6) which implies that 퐷퐹P(푦, 푝) has dense image whenever 푦 ∈ 푃 ∖ 퐴.

For the third property, let us investigate the derivative of 퐹 .

퐷푃 퐹(푦,푝)(푣) = 퐷푋 Θ(휎(푦),훾(푦)푝) ∘ 퐷휎푦(푣) + 퐷PΘ(휎(푦),훾(푦)푝)(퐷훾푦(푣) · 푝)

Due to Axiom (P4), 퐷푋 Θ ∈ GL푟푒푠. This together with 퐷휎 being negative map implies 퐷푋 Θ(휎(푦),훾(푦)푝) ∘ 퐷휎푦 : 푇 푃 → 푇 푋 is a negative map. The operator 푣 ↦→

퐷PΘ(휎(푦),훾(푦)푝)(퐷훾푦(푣) · 푝) is of rank one since 퐷훾푦 : 푇푦푃 → R is just a linear functional. Combining these we get that 퐷푃 퐹(푦,푝) is a negative map. Furthermore, Axiom

(P5) implies that 퐷P퐹(푦,푝) is compact.

퐾 Finally, we turn to proving that 휎 = 퐹 |푃 ×퐾 : 푃 × 퐾 → 푋 is a chain for any compact manifold with corners 퐾 ⊂ P. Firstly, 휎퐾 is a smooth map. Since

퐾 퐾 퐷푃 휎(푦,푝) = 퐷푃 퐹(푦,푝) we conclude that 퐷푃 휎(푦,푝) is a negative map. The operator 퐷휎퐾 퐾 differs from 퐷푃 휎퐾 by a finite rank operator, thus it follows that 퐷휎(푦,푝) is a negative map for any (푦, 푝) ∈ 푃 × 퐾.

To prove Axiom (M1) notice that we can write

ℒ(휎퐾 (푦, 푝)) = [ℒ(Θ(휎(푦), 훾(휎(푦)) · 푝)) − ℒ(휎(푦))] + ℒ(푦).

By Axiom (P2) we obtain that

|ℒ(Θ(휎(푦), 훾(휎(푦)) · 푝)) − ℒ(휎(푦))| ≤ 훾(휎(푦)) ·‖푝‖· 푙(‖휎(푦)‖2) = 휂(휎(푦))‖푝‖ ≤ 1 (3.1)

퐾 퐾 and therefore ℒ(휎 (푦, 푝)) is bounded above. Suppose 휎 (푦푖, 푝푖) = Θ(휎(푦푖), 훾(휎(푦푖))푝푖) ⇀

푥˜. We can choose a subsequence such that 푝푖 → 푝 and 훾(푦푖) → 푐, and thus

훾(휎(푦푖))푝푖 → 푐푝. By Axiom (P1) we get 휎(푦푖) ⇀ 푥, and therefore by Axiom (P3) we

63 퐾 obtain ℒ(휎 (푦푖, 푝푖)) − ℒ(휎(푦푖)) → ℒ(˜푥) − ℒ(푥). Finally

퐾 퐾 lim sup ℒ(휎 (푦푖, 푝푖)) = lim sup ℒ(휎(푦푖)) + lim(ℒ(휎 (푦푖, 푝푖)) − ℒ(휎(푦푖)) ≤ ℒ(푥) + ℒ(˜푥) − ℒ(푥)

= ℒ(˜푥) which shows that ℒ is weakly upper semi-continuous on the weak closure of im 휎퐾 .

퐾 Suppose ℒ(휎 (푦푖, 푝푖)) is bounded. By Equation 3.1 we get that ℒ(휎(푦푖)) is bounded and therefore 휎(푦푖) has a weakly convergent subsequence. Restricting further to a subsequence such that 푝푖 → 푝 and 훾(휎(푦푖)) → 푐 by Axiom (P1) we get that 퐾 휎 (푦푖, 푝푖) = Θ(푦푖, 푐푖푝푖) is weakly convergent.

퐾 퐾 Finally suppose 휎 (푦푖, 푝푖) ⇀ 푥˜ and ℒ(휎 (푦푖, 푝푖)) → ℒ(˜푥). Again, choosing a subsequence such that 푝푖 → 푝 and 훾(휎(푦푖)) → 푐, using Axiom (P3) we deduce that

퐾 퐾 lim ℒ(휎(푦푖)) = lim ℒ(휎 (푦푖, 푝푖)) − lim(ℒ(휎 (푦푖, 푝푖)) − ℒ(휎(푦푖)) = ℒ(˜푥) − ℒ(˜푥) + ℒ(푥)

= ℒ(푥)

where 푥 is the weak limit of 휎(푦푖) given by Axiom (P1). It follows that 휎(푦푖) → 푥 퐾 and therefore 휎 (푦푖, 푝푖) = Θ(휎(푦푖), 훾(휎(푦푖))푝푖) → 푥 = Θ(푥, 푐푝).

There is a version of this theorem for the adapted case.

Theorem 2. Let 퐺 be a correspondence from X to Y and Θ be a family of 퐺- adapted admissible perturbations. Suppose we are given a chain 휎 : 푃 → X and

a closed subset 퐴 ⊂ 푃 such that there exists a smooth function 휂 : 푃 → R such

that 휂|퐴 ≡ 0 and for any 푦 ∈ 푃 ∖ 퐴 we have 휂(푝) ∈ (0, 1]. Then there is a smooth

map 퐹 = 퐹휎,Θ : P × 푃 → 푋 such that:

1. For any (푦, 푝) ∈ 푃 × {0} ∪ 퐴 × P we have 퐹 (푦, 푝) = 휎(푦).

64 푠 2. For each 푝 ∈ 푃 ∖ 퐴 the restriction 퐹 |{푝}×P is transverse to 퐺 .

3. At each point (푦, 푝) ∈ 푃 ×P, the differential 퐷푃 퐹(푦,푝) is a negative map and

퐷P퐹(푦,푝) is compact.

4. If 퐾 ⊂ P is a compact manifold with corners (in particular, finite dimen-

sional), then 휎퐾 = 퐹 |퐾×푃 is a chain in X.

Proof. The proof is identical to the proof of Theorem 1.

Lemma 3.19. In the setting of Theorem 1 or Theorem 2, the index of the perturbed chain equals ind 퐹 |퐾×푃 = ind 휎 + dim 퐾.

Finally, we conclude that we can indeed achieve the desired transversality in a way that preserves the homology classes of cycles.

Theorem 3. Let 휎 : 푃 → X and 휏 : 푄 → X∨ be chains and assume there exists a family of admissible perturbations for X. Then there exists 휎′ : 푃 → X

′ transverse to 휏 and Σ: 푃 × [0, 1] → X such that Σ|푃 ×{0} = 휎 and Σ|푃 ×{1} = 휎 . Furthermore, for any two such perturbations 휎′ and 휎′′ there exists a chain Σ′ : 푃˜ → X which is transverse to 휏 and 휕Σ′ = 휎′ − 휎′′.

Proof. Let 퐹 = 퐹휎,P be constructed as in Theorem 1. Take any (푦, 푝) ∈ 푃 × P and

푧 ∈ 푍 such that 퐹 (푦, 푝) = 휏(푧), and define 푥 = 퐹 (푦, 푝) = 휏(푧). Then 퐷푃 퐹(푦,푝) is a negative map while 퐷휏푧 is a positive map, thus 퐷푃 퐹(푦,푝) + 퐷휏푧 : 푇푦푃 ⊕ 푇푧푄 → 푇푥푋 is Fredholm, in particular its image is of finite codimension and closed. Therefore

퐷퐹(푦,푝) +퐷휏푧 : 푇푦푃 ⊕푇푝P⊕푇푧푄 → 푇푥푋 also has closed image, thus from the density of 퐷퐹 we deduce it is onto. So 퐹 t 휏, or equivalently 퐹 × 휏 : 푃 × P × 푄 → 푋 × 푋 is transverse to the diagonal Δ ⊂ 푋 × 푋. It follows that (퐹 × 휏)−1(Δ) ⊂ 푃 × P × 푄 is a submanifold.

65 Recall that 퐷푃 퐹(푦,푝) + 퐷휏푧 : 푇푦푃 ⊕ 푇푧푄 → 푇푥푋 is Fredholm. It follows that

퐷푃 퐹(푦,푝) ⊕ 퐷휏푧 + id푇(푥,푥)Δ : 푇푦푃 ⊕ 푇푧푄 ⊕ 푇(푥,푥)Δ → 푇푥푋 ⊕ 푇푥푋 is also Fredholm, and thus (퐷푃 퐹(푦,푝)⊕퐷휏푧+id푇(푥,푥)Δ)⊕id푇푝P : 푇푦푃 ⊕푇푧푄⊕푇(푥,푥)Δ⊕푇푝P → 푇푥푋⊕푇푥푋⊕푇푝P is Fredholm as well. The operator 퐷P퐹 is compact and therefore the full derivative of the map (퐹 ×휏 +idΔ)×idP : 푃 ×P×푄 → 푋 ×푋 ×P is Fredholm. By Proposition 2.42 it follows that the projection (퐹 × 휏)−1(Δ) → P is Fredholm. By Theorem 17 it follows that 퐹 (·, 푝) × 휏(·): 푃 × 푄 → 푋 × 푋 is transverse to Δ for any 푝 in a residual subset of P.

′ The chain 휎 (·) = 퐹휎,P(·, 푝) is transverse to 휏 and cobordant to 휎 via 퐹 |푃 ×퐾 where ′ 퐾 = [0, 1] · 푝 ⊂ P. Also notice that 휕휎 (·) = 퐹휕휎,P(·, 푝) and therefore 휕휎 = 0 implies 휕휎′ = 0.

′ ′′ ˜ ′ ′′ Finally, if we have 휕Σ = 휎 − 휎 , Σ: 푃 → 푋, with both 휎 t 휏 and 휎 t 휏, then we can choose a smooth function 휂 : 푃˜ → R which is zero on boundary components of Σ belonging to 휎′ or 휎′′ (the sum of which we denote by 퐴 ⊂ 푃˜) and nonzero everywhere else, and bounded by 1. Theorem 1 applied to these 퐴 and 휂 implies the

′ ′ ′ ′′ existence of a chain Σ (·) = 퐹Σ,P(·, 푝) transverse to 휏 with 휕Σ = 휎 − 휎 .

Theorem 4. Let 휎 : 푃 → 푋 be a cycle and 퐺 : 푍 → 푋∨ ×푌 be a correspondence and assume there exists a family of 퐺-adapted perturbations. Then there exists

′ 푠 휎 : 푃 → X transverse to 퐺 and Σ: 푃 × [0, 1] → X such that Σ|푃 ×{0} = 휎 and ′ ′ ′′ Σ|푃 ×{1} = 휎 . Furthermore, for any two such perturbations 휎 and 휎 there exists a chain Σ′ : 푃˜ → X which is transverse to 퐺푠 and 휕Σ′ = 휎′ − 휎′′.

Proof. The proof follows the argument of the proof above, except that instead of 휏 we

푠 푠 consider 퐺 and the map 퐹휎,P ×퐺 is not necessarily Fredholm, but is right Fredholm (i.e., has closed image and finite dimensional cokernel) (c.f., proof of Proposition

2.40).

66 3.4 Suspension isomorphisms

In this section we aim to prove that suspending a Floer space X by a (possibly finite- dimensional) Hilbert space with a “nonnegative” or “negative” functional (and thus by products of such spaces) induces isomorphisms on semi-infinite homology. We expect these results to eventually be generalized to prove a Künneth formula for a product as well as Thom isomorphisms for Hilbert bundles.

Proposition 3.20. Let 퐻 be any Hilbert space (possibly finite-dimensional). Let

ℒ : 퐻 → R be bounded below and weakly lower semi-continuous, and the polarization be given by 풥 be induced by 푇 −퐻 = {0}. Denote the Floer space by H = (퐻, ℒ, 풥 ).

Then 퐻퐹푘(H) ≃ 퐻퐹푘({0}) for every 푘 ∈ Z.

Proof. We could follow the argument of Proposition 3.15, i.e. notice that 휎 is a chain if and only if it is finite-dimensional and compact. However, we provide a different proof to set the stage for further proofs in this section. This characterization of chains shows that without loss of generality we can assume ℒ(ℎ) = ‖ℎ‖2. Notice that in this case the inclusion 푖 : {0} → 퐻 and the projection 휋 : 퐻 → {0}

have graphs which are Floer correspondences, i.e., Γ푖 is a Floer correspondence from

the trivial Floer space O to H and Γ휋 a Floer correspondence from H to O, both of index 0 (cf. Proposition (2.86). We claim that the isomorphism in the proposition is

induced by these Floer correspondences. On one hand we have 퐻퐹*(Γ휋) ∘ 퐻퐹*(Γ푖) =

퐻퐹*(Γ휋 ∘ Γ푖) = 퐻퐹*(Γid{0} ) = id퐻퐹*({0}). On the other hand 퐻퐹*(Γ푖) ∘ 퐻퐹*(Γ휋) =

퐻퐹*(Γ0) where 0 : 퐻 → 퐻 is given by ℎ ↦→ 0 · ℎ = 0. However, the graph Γ푀 of the map 푀 : 퐻 × [0, 1] → 퐻 given by 푀(ℎ, 푡) = 푡 · ℎ is a dense parametrized

Floer correspondence due to Proposition 2.86. Since 푀1 = 푀|퐻×{1} is isomorphic to the identity map and 푀0 = 푀|퐻×{0} is isomorphic to the zero map, thus the map

퐶퐹*(Γ푀 ): 퐶퐹*(퐻) → 퐶퐹*(퐻) is a homotopy between 퐶퐹*(Γ0) and 퐶퐹*(Γid):

휕퐶퐹*(Γ푀 )(휎) − 퐶퐹*(Γ푀 )(휕휎) = 퐶퐹*(Γ휕푀 )(휎)

= 퐶퐹*(Γid)(휎) − 퐶퐹*(Γ0)(휎)

67 which finishes the proof.

The proof readily generalizes to prove a suspension isomorphism for a non-negative definite quadratic functional.

Proposition 3.21. Let X = (푋, ℒ푋 , Π푋 ) be any Floer space. Let 퐻 be a Hilbert

space with ℒ퐻 (ℎ) = ⟨퐴ℎ, ℎ⟩ for some nonnegative self-adjoint 퐴 ≥ 0 and polarization − induced by 푇 퐻 = {0}, giving a Floer space H = (퐻, ℒ퐻 , 풥퐻 ). Then the inclusion

푖 : 푋 ≃ 푋 × {0} ˓→ 푋 × 퐻 induces isomorphisms 퐻퐹푘(X) ≃ 퐻퐹푘(X × H).

Proof. Due to Proposition 2.86, the graphs of 푖 : 푋 → 푋 × 퐻 and 휋푋 : 푋 × 퐻 → 푋 are dense Floer correspondences, and by Lemma 2.87 they have index zero.

Moreover 휋푋 ∘ 푖 = id. Therefore it suffices to prove 푖 ∘ 휋푋 is homotopic to identity through a homotopy which graph is a Floer correspondence. This is achieved using the homotopy 퐺 : 푋 × 퐻 × [0, 1] → 푋 × 퐻, 퐺(푥, ℎ, 푡) = (푥, 푡 · ℎ), the graph of which

is again a dense Floer correspondence due to Proposition 2.86.

The “negative” case is slightly harder to prove since it involves using non-dense correspondences and performing a blow-up along the “zero locus”.

Proposition 3.22. Let 퐻 be any Hilbert space (possibly finite-dimensional) and as-

sume that for a Floer space H = (퐻, ℒ퐻 , 풥퐻 ) the map id : 퐻 → 퐻 is a chain of index

0. Then 퐻퐹푘(H) ≃ 퐻퐹푘({0}) for any 푘 ∈ Z.

Proof. Let us start by noticing that in this case 휎 : 푃 → 퐻 is a chain if and only if 퐷휎 is Fredholm (since 푇 −퐻 = 푇 퐻 by the index 0 assumption) and 휎 is proper. These conditions do not depend on the exact values of ℒ, just on the condition that id : 퐻 → 퐻 is a chain of index 0, therefore we can take ℒ(푥) = −‖푥‖2 for simplicity. Notice that −ℒ satisfies the hypotheses of Proposition (3.20). The proof here will be indeed dual to the proof of that Proposition besides that here we will need to construct a family of admissible perturbations.

∨ Firstly, the dual graph Γ푖 of the inclusion 푖 : {0} ˓→ 퐻 is a Floer correspondence ∨ from H to O, and the dual graph Γ휋 of the projection 휋 : 퐻 → {0} is a dense Floer

68 correspondence from O to H, both of which are easily checked. Also, by Lemma 2.87 they each have index zero as Floer correspondences. We start by showing that there

∨ exists a Γ푖 -adapted family of perturbations. In this case we can actually construct a family of admissible perturbations for H = (퐻, ℒ, Π). Take any Hilbert space 퐻′ with a compact dense linear operator 휄 : 퐻′ → 퐻. We claim that Θ(ℎ, 푝) = ℎ + 휄(푝) is a family of admissible perturbations:

• Axiom (P1) follows immediately.

• Axiom (P2) is satisfied since |ℒ(ℎ+휄(푝))−ℒ(ℎ)| ≤ 2‖휄‖·‖푝‖·‖ℎ‖+‖휄‖2 ·‖푝‖2 ≤ ‖푝‖(2‖휄‖‖ℎ‖ + ‖휄‖2).

• Axiom (P3) is satisfied since

ℒ(ℎ + 휄(푝)) − ℒ(ℎ) = 2⟨휄(푝), ℎ⟩ + ‖휄(푝)‖2

which is weakly continuous in both 푝 and ℎ, simultaneously.

• Axiom (P4) follows immediately since 퐷푋 Θ = id푇 퐻 .

• Axioms (P5) and (P6) follow immediately since 퐷PΘ = 휄 is compact and dense.

∨ ∨ We claim that the maps on homology induced by correspondences Γ푖 and Γ휋 give ∨ ∨ the required isomorphism. While Γ푖 ∘Γ휋 is just the identity correpondece on {0} and ∨ ∨ ∨ ∨ thus 퐻퐹*(Γ푖 ) ∘ 퐻퐹*(Γ휋 ) = id, we need to prove that Γ휋 ∘ Γ푖 , which sends a chain −1 휎 t {0} to the chain 휎 (0) × 퐻 → 퐻, (푝, ℎ) ↦→ ℎ, induces identity on homology. We will actually show that it is homotopic to identity.

We would like to use the correspondence given by the graph Γ퐹 of 퐹 : 퐻×[1, ∞) →

퐻, 퐹 (ℎ, 푠) = 푠·ℎ, to “flow the cycles to infinity”. However, Γ퐹 is not a correspondence as defined. There arises the problem of “completing” or “compactifying” itas 푠 → ∞. We will do it by employing a suitable reparametrization.

∨ ∨ Let us define 퐺 : 퐻 ×[0, 1] → 퐻 via 퐺(ℎ, 푡) = 푡·ℎ so that Γ퐺 : 퐻 ×[0, 1] → 퐻 ×퐻 ∨ ∨ is given by Γ퐺(ℎ, 푡) = (푡 · ℎ, ℎ). Notice that Γ퐺|퐻×(0,1] is isomorphic to Γ퐹 via the 1 ∨ substitution 푠 = 푡 . This Γ퐺 is indeed a correspondence:

69 ∨ • Axiom (C1). Γ퐺 is weakly continuous, has weakly closed image and ℒ(ℎ) − ℒ(푡ℎ) = (푡2 − 1)‖ℎ‖2 is bounded above by 0 and weakly upper semi-continuous on 퐻 × [0, 1].

2 • Axiom (C2). If ℒ(ℎ푖) = −‖ℎ푖‖ is bounded below, then {(ℎ푖, 푡푖)}푖 is weakly precompact.

′ • Axiom (C3). If 푡푖ℎ푖 → ℎ then either 푡푖 → 푡 and thus ℎ푖 → ℎ, or 푡푖 → 0, and 2 2 then given ℎ푖 ⇀ ℎ and ℒ(ℎ푖) − ℒ(푡푖ℎ푖) → ℒ(ℎ) we obtain ‖ℎ푖‖ → ‖ℎ‖ and

thus ℎ푖 → ℎ.

− + − ∨ 푡 • Axiom (P1). 푇 퐻 = 푇 퐻 and 푇 퐻 = 0, so Π 퐷퐻 Γ퐺 = id which is Fredholm, − ∨ 푡 thus Π 퐷Γ퐺 , which differs from the latter map by a finite-dimensional map, is also Fredholm. Also 푇 +퐻 = 0 so any map into 푇 +퐻 is compact.

∨ ∨ By construction, Γ퐺1 = Γ퐺|퐻×{1} is isomorphic to the identity correspondence ∨ ∨ ∨ ∨ ∨ while Γ퐺0 = Γ퐺|퐻×{0} is isomorphic to the composition Γ휋 ∘ Γ푖 . Moreover, Γ퐺|퐻×(0,1] ∨ ∨ is dense, and thus any Γ푖 -adapted family of perturbations is also Γ퐺-adapted. There- ∨ ∨ ∨ ⊥Γ퐺 ⊥Γ퐺 fore taking the fiber product with 퐺 gives a map 퐶퐹*(Γ퐺): 퐶퐹* (H) → 퐶퐹* (H) ∨ ∨ ∨ which satisifes 휕 ∘ 퐶퐹*(Γ퐺)(휎) − 퐶퐹* ∘ 휕(휎) = 퐶퐹*(휕Γ퐺)(휎) = 퐶퐹*(Γ퐺1 )(휎) − ∨ ∨ ∨ ∨ ∨ 퐶퐹*(Γ퐺0 )(휎) = 휎 − 퐶퐹*(Γ휋 ∘ Γ푖 )(휎), thus showing 퐶퐹*(Γ휋 ∘ Γ푖 ) is homotopic to the identity as a chain map.

Remark 3.23. Notice that if im 휎 ⊂ 퐻 ∖ {0} then the correspondence 퐺 “flows” the whole cycle 휎 “to infinity”, and thus it is nullhomologous. For 푘 > 0 any cycle 휎 is negligible, while for 푘 < 0 it can be perturbed to avoid {0}. Finally, for 푘 = 0 it can be perturbed to be transverse to {0} and have 휎−1(0) discrete. Then [휎] is sent to

#휎−1(0) ∈ F.

Proposition 3.24. Let X = (푋, ℒ푋 , Π푋 ) be any Floer space. Let 퐻 be a Hilbert

space with ℒ퐻 (ℎ) = ⟨퐴ℎ, ℎ⟩ for some self-adjoint negative definite 퐴 and polarization − induced by 푇 퐻 = 퐻, giving a Floer space H = (퐻, ℒ퐻 , 풥퐻 ). Then the dual graph Γ∨ of the map 휋 : 푋 × 퐻 → 푋 is a dense Floer correspondence which induces 휋푋 푋 isomorphisms 퐻퐹 (Γ∨ ): 퐻퐹 (X) → 퐻퐹 (X × H) for all 푘 ∈ . 푘 휋푋 푘 푘 Z

70 ∨ Proof. That the dual graph Γ푖 of the inclusion 푖 : 푋 → 푋 × 퐻 is a Floer correspondence and that the dual graph Γ∨ of the projection 휋 : 푋 × 퐻 → 푋 is a dense 휋푋 푋 Floer correspondence are both straightforward to check. Also, by Lemma 2.87 they each have index zero.

∨ We first claim that there isa Γ푖 -adapted family of perturbations. Indeed, let P ⊂ 퐻′ be an open unit ball and 휄 : 퐻′ → 퐻 be any compact map, then Θ:

∨ 푋 × 퐻 × P → 푋 × 퐻 given by Θ(푥, ℎ, 푝) = (푥, ℎ + 휄(푝)) is a family of Γ푖 -adapted perturbations (cf. proof of Proposition 3.22). Secondly, notice that the composition Γ∨ ∘ Γ∨ = Γ∨ = Γ∨ = Γ is the 푖 휋푋 휋푋 ∘푖 id푋 id푋 identity correspondence and therefore induces the identity map on 퐶퐹*(X). The other composition 푆 = Γ∨ ∘ Γ∨ : 푋 × 퐻 → (푋 × 퐻) × (푋 × 퐻), 푆(푥, ℎ) = 휋푋 푖 ((푥, 0), (푥, ℎ)), is a correspondence, not a dense one, but Θ is also 푆-adapted. Thus it induces a map on semi-infinite homology and we want to prove that this map is identity. Indeed, notice that 푆 = Γ∨ . The map 퐺 : 푋 × 퐻 × [0, 1] → 푋 × 퐻 given by 푖∘휋푋

퐺(푥, ℎ, 푡) = (푥, 푡 · ℎ) has 퐺1 = 퐺|푋×퐻×{1} ≃ id푋×퐻 and 퐺0 = 퐺|푋×퐻×{0} ≃ 푖 ∘ 휋푋 . ∨ Therefore it suffices to show that the dual graph Γ퐺 of 퐺 is a Floer correspondence and thus yields a chain homotopy between id and 퐻퐹 (Γ∨ ) ∘ 퐻퐹 (Γ∨), finishing the * 휋푋 * 푖 proof.

∨ Indeed, Γ퐺 : 푋 × 퐻 × [0, 1] → (푋 × 퐻) × (푋 × 퐻) is given by (푥, ℎ, 푡) ↦→ ((푥, 푡ℎ), (푥, ℎ)).

• Axiom (C1) follows from the weak continuity of 퐺 and the fact that ℒ(푥, ℎ) − ℒ(푥, 푡ℎ) = (푡2 − 1)‖ℎ‖2 is weakly upper semicontinuous.

• Axiom (C2). If (푥푖, 푡푖ℎ푖) is weakly precompact and ℒ(푥푖, ℎ푖) is bounded below, 2 2 then ℒ(푥푖, 푡푖ℎ푖) is bounded and thus ℒ(푥푖, ℎ푖) − ℒ(푥푖, 푡푖ℎ푖) = (푡푖 − 1)‖ℎ푖‖ is

bounded below, implying ℎ푖 is weakly precompact. Thus (푥푖, ℎ푖) is weakly precompact.

′ • Axiom (C3). We assume that (푥푖, 푡푖ℎ푖) → (푥, ℎ ) and (푥푖, ℎ푖) ⇀ (푥, ℎ) and 2 2 ′ 2 2 (푡푖 − 1)‖ℎ푖‖ → ‖ℎ ‖ − ‖ℎ‖ . Without loss of generality, 푡푖 → 푡. If 푡 > 0, then

71 ′ 2 2 ′ 2 ℎ푖 → ℎ /푡. If 푡 = 0, then from the convergence (푡푖 − 1)‖ℎ푖‖ → ‖ℎ ‖ − ‖ℎ‖ it 2 2 ′ 2 ′ follows that ‖ℎ푖‖ → ‖ℎ‖ − ‖ℎ ‖ . Thus ℎ푖 is bounded and therefore ℎ = 0, 2 2 which implies ‖ℎ푖‖ → ‖ℎ‖ and therefore ℎ푖 → ℎ.

• Axiom (P1). Since 퐺 = id푋 ×퐺퐻 , we only need to check this for 퐺퐻 . Moreover, 푇 퐻+ = {0} and 푇 퐻− = 푇 퐻 and indeed 퐷−Γ∨ 푡 = id + 퐷− Γ∨ 푡 is 퐺퐻 푇 퐻 [0,1] 퐺퐻 Fredholm since the last summand is compact. Moreover 푇 +퐻 = {0} implies that 퐷+Γ∨ 푡 is compact. 퐺퐻

∨ This proves that Γ퐺 is a Floer correspondence and thus provides the desired homotopy.

∨ Remark 3.25. Notice that the correspondence Γ퐺 defined above is a compactification of the correspondence (푥, 푣, 푠) ↦→ ((푥, 푣), (푥, 푠푣)) for (푥, 푣, 푠) ∈ 푋 × 퐻 × [1, ∞) via

1 the change of variables 푠 = 푡 , 푣 = 푡ℎ, which is the gradient flow of ℒ퐻 .

3.5 Perturbing functionals

For defining Floer homology, one often needs to perturb the functional in orderto achieve regularity of moduli spaces, and show that the resulting homology does not depend on the perturbation. The perturbations used in [KM07] are very well-behaved and we will show that they actually do not change the set of chains altogether.

Proposition 3.26. Assume 푓 = ℒ′ − ℒ : 푋 → R is bounded and weakly continuous. Then 휎 : 푃 → 푋 is a semi-compact map for ℒ if and only if it is a semi-compact map for ℒ′.

Proof. Assume that 휎 is a semi-compact map for ℒ. We need to prove that it is a semi-compact map for ℒ′. Firstly, ℒ ∘ 휎 is bounded above and ℒ′ − ℒ is bounded, so ℒ′ ∘ 휎 is bounded above

72 as well. Take 휎(푥푖) ⇀ 푦. Then

′ lim sup ℒ (휎(푥푖)) = lim 푓(휎(푥푖)) + lim sup ℒ(휎(푥푖)) ≤ 푓(푦) + ℒ(푦)

= ℒ′(푦)

as required by Axiom M1.

′ Secondly, take any sequence 푥푖 and assume ℒ (휎(푥푖)) is bounded. Then ℒ(휎(푥푖)) = ′ ℒ (휎(푥푖)) − 푓(휎(푥푖)) is also bounded and therefore 휎(푥푖) is weakly precompact, as required by Axiom M2.

′ ′ ′ Finally, assume that 휎(푥푖) ⇀ 푦 and that ℒ (휎(푥푖)) → ℒ (푦). Since ℒ −ℒ is weakly

continuous we obtain that ℒ(휎(푥푖)) → ℒ(푦) which implies 휎(푥푖) → 푦, as required by Axiom M3.

Corollary 3.27. If ℒ′ − ℒ : 푋 → R is bounded and weakly continuous then the iden- ′ tity map induces an isomorphism of homology groups 퐻퐹푘(푋, ℒ, 풥 ) ≃ 퐻퐹푘(푋, ℒ , 풥 ) for any polarization 풥 .

′ Proposition 3.28. Assume 푓푖 = ℒ푖 − ℒ푖 : 푋푖 → R are bounded and weakly continu-

ous. Then 퐹 : 푍 → 푋1 × 푋2 is a semi-compact correspondence for ℒ1 and ℒ2 if and ′ ′ only if it is a semi-compact correspondence for ℒ1 and ℒ2.

′ ′ ′ Proof. Regarding Axiom C1, the equivalence follows since ℒ−ℒ = ℒ2 −ℒ1 −(ℒ2 −ℒ1) is bounded and weakly continuous.

푡 ′ 푡 Regarding Axiom C2, ℒ2(푓 (푧푖)) is bounded below if and only if ℒ2(푓 (푧푖)) is bounded below.

′ ′ Regarding Axiom C3, ℒ(푓(푧푖)) → ℒ(푥, 푦) if and only if ℒ (푓(푧푖)) → ℒ (푥, 푦) ′ ′ because weak continuity implies ℒ(푓(푧푖)) − ℒ (푓(푧푖)) → ℒ(푥, 푦) − ℒ (푥, 푦).

73 74 Chapter 4

Seiberg-Witten Floer Spaces

This chapter introduces the Seiberg-Witten Floer spaces for rational homology spheres 푌 with a spin푐 -structure s. For a 4-manifold 푋 with 휕푋 = 푌 and a spin푐 -structure s restricting to that on 푌 , the restriction map from the moduli space of solutions on 푋 to the configurations on 푌 yields a closed 푆1-equivariant chain in 푌 . Similarly, the moduli space of solutions on a cobordism 푊 between rational homology spheres 푌1 푐 1 and 푌2 with a chosen spin -structure gives a dense 푆 -equivariant correspondence. In this chapter, we establish these facts as well as prove that composing cobordisms 푊1 and 푊2 gives a correspondence which is (homotopic to) the composition of the respective correspondences. We show that there is a family of admissible perturbations which shows the existence of a Poincaré pairing. Finally, we prove the invariance of homology under perturbations of the Chern-Simons-Diract functional used in [KM07] to define Monopole homology .

4.1 Seiberg-Witten-Floer spaces and gauge actions

Here we introduce the Chern-Simons-Dirac functional and use it to define Seiberg- Witten-Floer spaces. We expect the (non-equivariant) semiinfinite homology of these spaces to be isomorphic to the reduced homology 퐻푀̃︂ *(푌, s) as defined by Bloom. Our future aim is to show that appropriate 푆1-equivariant semiinfinite homologies are isomorphic to the to, from and bar flavors of Monopole homology.

75 Let 푌 be a smooth, oriented, closed, connected, Riemannian 3-manifold with

푐 푏1(푌 ) = 0 and s be a spin structure on 푌 . Define the configuration space 풞(푌, s) to 2 푐 be the 퐿1/2 completion of the set of pairs (퐵, Ψ) where 퐵 is a spin connection and Ψ is a section of the spinor bundle 푆 (or simply, a spinor). We will also write

2 풞(푌, s) = 풜(푌, s) × 퐿1/2(푋; 푆)

2 푐 where 풜(푌, s) is the 퐿1/2 completion of the space of spin connections on 푆. 푐 Fix a smooth reference spin connection 퐵0. Define

표 * ℬ (푌, s) = {(퐵, Ψ)|푑 (퐵 − 퐵0) = 0} ⊂ 풞(푌, s) (4.1)

to be the subset of configurations with 퐵−퐵0 coclosed; this is also called the Coulomb slice of the configuration space (cf. [Kha13,Kha15], where it is denoted by Coul(푌 )). We will abbreviate ℬ표 = ℬ표(푌, s), and we will often identify

표 * 2 2 1 2 ℬ ≃ 푖 ker(푑퐿2(푌 ;Λ1)) ⊕ 퐿1/2(푌 ; 푆) ⊂ 퐿1/2(푌 ; 푖Λ ) ⊕ 퐿1/2(푌 ; 푆).

The tangent space 푇 ℬ표 is isomorphic, at every point 푝, to

표 * 2 푇푝ℬ ≃ 푖 ker(푑퐿2(푌 ;Λ1)) ⊕ 퐿1/2(푌 ; 푆).

+ 표 − 표 The operator 퐷 = (*푑, 퐷퐵0 ) is elliptic on this space. We define 푇 ℬ (resp. 푇 ℬ ) to be the positive (resp. nonpositive) eigenspace of that operator. We denote the

corresponding polarizaiton as 풥 = 풥(푌,s).

The Chern-Simons-Dirac functional ℒ = ℒ(푌,s) is defined by

∫︁ ∫︁ 1 푡 푡 1 ℒ(퐵, Ψ) = − (퐵 − 퐵0) ∧ (퐹퐵푡 + 퐹퐵푡 ) + ⟨퐷퐵Ψ, Ψ⟩, (4.2) 8 푌 0 2 푌

Definition 4.1 (Seiberg-Witten-Floer space). We call the triple B(푌, s) =

표 (ℬ (푌, s), ℒ(푌,s), 풥(푌,s)) the Seiberg-Witten-Floer space.

76 Theorem 5. B(푌, s) is a Floer space.

Proof. Clearly, ℬ표 is a Hilbert space, 푇 ±ℬ표 are smooth subbundles of 푇 ℬ표 and ℒ is continuous, so it suffices to show that Axiom F1 holds. However, from the definition of ℒ, we have

2 |ℒ(퐵, Ψ)| ≤ 퐶(‖(퐵 − 퐵0, Ψ)‖퐿2 + ‖(퐵 − 퐵0, Ψ)‖ 2 ). 1/2 퐿1/2

Any weakly precompact set 푆 in B is bounded, thus, by the inequality above, ℒ is

bounded on 푆.

Remark 4.2. We can canonically identify spin푐 structures on 푌 and −푌 ; thus, having

푐 chosen a metric on 푌 and a spin connection 퐵0, we can canonically identify the 표 표 spaces ℬ (푌, s) ≃ ℬ (−푌, s) and in such case we have ℒ(푌,s) = −ℒ(−푌,s) and the ±1

eigenspaces of (*푑, 퐷퐵0 ) are switched. What follows is that as a Floer space B(푌, s) is isomorphic to (B(−푌, s))∨.

There gauge group

2 1 풢(푌 ) = {푢 ∈ 퐿3/2(푌 ; C)|푢(푥) ∈ 푆 almost everywhere} (4.3)

acts on 풞(푌, s) via 푢(퐵, Ψ) = (퐵 − 푢−1푑푢, 푢Ψ) (4.4)

leaving ℒ unchanged.

2 2 1 If we took configurations in 퐿푠 and gauge transformations in 퐿푠+1 for 푠 > 2 , then Theorem 12 would prove that this action is well-defined and continuous, while Lemma

1 A.4 would additionally imply that it is smooth. Neither is the case for 푠 = 2 . Proposition 4.3. The gauge action 풢(푌 ) × 풞(푌, s) → 풞(푌, s) is continuous and

2 ∞ smooth for the 퐿3/2 ∩ 퐿 topology on 풢(푌 ) (i.e., topology induced by the norm ‖·

‖ 2 ∞ = ‖·‖ 2 + ‖·‖퐿∞ ). 퐿3/2∩퐿 퐿3/2

77 ∞ 2 Proof. Notice that 퐿 ∩ 퐿3/2 is a Banach space and 풢(푌 ) is its Banach submanifold. Now continuity follows from Proposition A.1 and smoothness from Lemma A.4.

Lemma 4.4. Any configuration (퐵, Ψ) may be put into the Coulomb gauge by a unique

2 푢 ∈ 풢(푌 ). This 푢 depends smoothly on 퐵 as an element of 퐿3/2.

1 * Proof. Since 푏1(푌 ) = 0, we have Ω (푌 ) = im 푑 ⊕ im 푑 , and for Sobolev spaces 2 1 2 * 2 2 * 푒푣푒푛 퐿1/2(푌 ; 푖Λ ) = 푑(퐿3/2(푌 ; 푖R))⊕푑 (퐿3/2(푌 ; 푖Λ )). Moreover, since 푑+푑 :Ω (푌 ) → 표푑푑 (︁ )︁ Ω (푌 ) is elliptic, with ker 푑|Ω0(푌 ) being the constant functions, we get that 푑 : 2 2 1 2 퐿3/2(푌 ; 푖R)0 → im 푑 ⊂ 퐿1/2(푌 ; 푖Λ ) is an isomorphism, where 퐿3/2(푌 ; 푖R)0 = {푢 ∈ 2 ∫︀ 퐿3/2(푌 ; 푖R)| 푌 푢 푑vol = 0}. Denote its inverse by 퐺푑. 2 2 * Let Π푑 : 퐿3/2(푌 ; 푖R) → 퐿3/2(푌 ; 푖R) be the projection onto im 푑 along im 푑 . Take * −1 * 푢 = exp(−퐺푑Π푑(퐵−퐵0)). Then 푑 ((퐵−푢 푑푢)−퐵0) = 푑 (퐵−퐵0−푑퐺푑Π푑(퐵−퐵0)) = * 푑 (퐵 − 퐵0 − Π푑(퐵 − 퐵0)) = 0, as wished.

The projection 풞(푌, s) → ℬ표(푌, s) is not continuous since the choice of 푢 is not continuous with respect to 퐿∞ norm on 풢(푌 ). However, since the action of a single element 푢 ∈ 풢(푌 ) is smooth, we can conclude that choosing the gauge slice “doesn’t matter” anyway:

표 ˜표 ˜ Corollary 4.5. Let ℬ and ℬ be obtained using reference connections 퐵0, 퐵0, respec- 표 ˜표 표 tively. Then the map 휑퐵0,퐵˜0 : ℬ → ℬ given by mapping a configuration (퐵, Ψ) ∈ ℬ to the unique representative (퐵,˜ Ψ)˜ ∈ ℬ˜표 of the same gauge orbit is a diffeomorphism. Moreover, it preserves the functional ℒ and the respective polarizations 풥 , 풥˜.

Proof. There is a unique element 푢퐵0,퐵˜0 ∈ 풢(푌 ) which transforms 퐵0 to a connection ˜ * 표 퐵0 + 푏 with 푑 푏 = 0. Action by this element on ℬ gives the desired diffeomorphism.

The differential of this diffeomorphism intertwines the operators (*푑, 퐷퐵0 ) and

(*푑, 퐷퐵˜0+푏), and the latter one differs by a compact operator Ψ ↦→ 휌(푏)Ψ (compact 2 2 * as a map 퐿 → 퐿 by Theorem 12) from (*푑, 퐷 ˜ ). Thus 휑 (풥 ) = 풥˜, as 1/2 −1/2 퐵0 퐵0,퐵˜0 wished.

78 4.2 Moduli spaces are cycles

푐 Suppose we have a 4-manifold 푋 such that 휕푋 = 푌 , with a spin structure s푋 restricting to s on 푌 . In this seciton we prove that the restriction map from the moduli space of solutions to Seiberg-Witten equations on 푋 in the double Coulomb gauge (ℳ표(푋, s)) to the space of configurations on 푌 in Coulomb gauge (ℬ(푌, s)) is a chain in the Floer space B(푌, s). We denote both by s and the spinor bundle on 푋 by 푆+ ⊕ 푆−. Note that over 푌 ,

+ − 2 there are identifications 푆 |푌 ≃ 푆 ≃ 푆 |푌 . Let 풜(푋, s) be the 퐿1 completion of the space of spin푐 connections on 푋 and define the space of configurations by

2 + 풞(푋, s) = 풜(푋, s) ⊕ 퐿1(푋; 푆 ).

Fix a smooth reference connection 퐴0. We define

표 * * ℬ (푋, s) = {(퐴, Φ) ∈ 풞(푋, s)|푑푋 (퐴 − 퐴0) = 0, 푑푌 (퐵 − 퐵0) = 0} (4.5) to be the configurations in the double Coulomb gauge (cf. CoulCC(푋) in [Kha13, Kha15]). The gauge group

{︁ ⃒ }︁ 2 ⃒ 1 풢(푋) = 푢 ∈ 퐿2(푋, C)⃒푢 : 푋 → 푆 almost everywhere (4.6) acts on 풞(푋, s) via 푢(퐴, Φ) = (퐴 − 푢−1푑푢, 푢Φ). (4.7)

Similarly to the 3-dimensional case we have:

Proposition 4.6. The gauge action 풢(푋) × 풞(푋, s) → 풞(푋, s) is well-defined. It is

2 ∞ continuous and smooth for the 퐿2 ∩ 퐿 topology on 풢(푋).

Lemma 4.7. Any configuration (퐴, Φ) can be put in the double Coulomb gauge by some 푢 ∈ 풢(푋). Moreover, one can fix such a smooth assignment 퐴 ↦→ 푢(퐴).

79 Proof. Let 푢 = 푒푓 , so that 푢−1푑푢 = 푑푓. We want

⎧ ⎪ * −1 ⎨⎪푑 (퐴 − 퐴0 − 푢 푑푢) = 0,

⎪ * −1 ⎩⎪푑 (퐵 − 퐵0 − 푢 푑푌 푢) = 0.

Equivalently, ⎧ ⎪ * * ⎨⎪푑 푑푓 = 푑 (퐴 − 퐴0),

⎪ * * ⎩푑푌 푑푌 푓 = 푑 (퐵 − 퐵0).

* 2 Since 푌 is a homology sphere, we have 퐵 − 퐵0 = 푑푌 ℎ + 푑푌 휔 for some ℎ, 휔 ∈ 퐿3/2 2 (recall that 퐵 − 퐵0 ∈ 퐿1/2). Precisely, ℎ = 퐺푑Π푑(퐵 − 퐵0) (c.f. proof of Lemma 4.4). Therefore, it suffices to solve for 푓 such that

⎧ ⎪ * * ⎨⎪푑 푑푓 = 푑 (퐴 − 퐴0), ⎪ ⎩⎪푓|푌 = ℎ.

2 This is the Dirichlet problem and thus it has a solution 푓 ∈ 퐿2 depending continuously 2 2 (and linearly) on (퐴−퐴0, ℎ) ∈ 퐿1×퐿3/2. Since ℎ = 퐺푑Π푑(퐵−퐵0), thus 푓 is continuous 2 푓 and linear in 퐴 − 퐴0 ∈ 퐿1. Thus 푢 = 푒 depends smoothly on 퐴 − 퐴0.

This does not give a smooth map 풞(푋, s) → ℬ표(푊, s) since we this choice is not continuous with respecto to the 퐿∞-topology on 풢(푋).

Definition 4.8 (moduli space on a manifold with boundary). Assume

휕푋 = 푌 has one connected component. Denote by ℛ : 풞(푋, s) → 풞(푌, s|푌 ) the 표 표 restriction map ℛ(퐴, Φ) = (퐴|푌 , Φ|푌 ),. Note that ℛ(ℬ (푋, s)) ⊂ ℬ (푌, s|푌 ). Define

{︁ ⃒ }︁ 표 표 ⃒ ℎ ℳ (푋, s) = (퐴, Φ) ∈ ℬ (푋, s)⃒SW(퐴, Φ) = 0 풢 (푋, 휕푋) ,

the Seiberg-Witten moduli space, i.e. the space of solutions to the Seiberg-Witten

80 equations defined by

2 − SW : 풞(푋, s) → 푖Ω+(푋)퐿2 ⊕ Γ(푆 )퐿2 (︂ )︂ 1 + −1 * + SW(퐴, Φ) = 퐹 푡 − 휌 ((ΦΦ ) ), 퐷 Φ (4.8) 2 퐴 0 퐴

modulo the gauge group

{︁ ⃒ }︁ ℎ 1⃒ * −1 1 풢 (푋, 휕푋) = 푢 : 푋 → 푆 ⃒푑푋 (푢 푑푢) = 0, 푢|휕푋 = 1 ≃ 퐻 (푋; Z).

In Floer theories, it is typical that such a manifold 푋 defines a relative invariant Ψ(푋) in the Floer theory of 푌 via the use of a space of solutions to some partial differential equations on 푋. Thus, it is natural to ask for the restriction map to define a cycle in the Floer space:

Theorem 6 (Seiberg-Witten moduli spaces are cycles). ℳ표(푋, s) is

표 a Hilbert manifold without boundary, and the restriction map ℛ(푋,s) : ℳ (푋, s) → 표 1 표 ℬ (푌, s|푌 ) is an 푆 -equivariant cycle in ℬ (푌, s|푌 ).

The circle 푆1 ⊂ C acts on configurations by multiplication 푧 :(퐴, Φ) ↦→ (퐴, 푧Φ), so the restriction map is 푆1-equivariant. Before we carry out the proof, we recall some definitions and results from [KM07]. Firstly, for (퐴, Φ) ∈ 풞(푋, s), we define the topological and analytic energies [KM07, Definition 4.5.4]:

∫︁ ∫︁ ∫︁ top 1 2 ℰ (퐴, Φ) = 퐹퐴푡 ∧ 퐹퐴푡 − ⟨Φ|푌 , 퐷퐵Φ|푌 ⟩ + (퐻/2)|Φ| , (4.9) 4 푋 푌 푌 ∫︁ ∫︁ ∫︁ ∫︁ 2 an 1 2 2 1 2 2 푠 ℰ (퐴, Φ) = |퐹퐴푡 | + |∇퐴Φ| + (|Φ| + (푠/2)) − . (4.10) 4 푋 푋 4 푋 푋 16

81 Here 퐵 = 퐴|푌 , 퐻 is the mean curvature of the boundary 푌 (which is zero if there is a cylindrical neighborhood of 푌 ), 푠 is the scalar curvature of 푋. We also have [KM07, Proposition 4.5.2]:

ℰ an(퐴, Φ) = ℰ top(퐴, Φ) + ‖ SW(퐴, Φ)‖2.

Therefore, whenever (퐴, Φ) solves the Seiberg-Witten equation, we will use the notation ℰ(퐴, Φ) do denote both the topological and the analytic energy.

Importantly, if (퐵, Ψ) = (퐴, Φ)|푌 , then

ℰ top(퐴, Φ) = −2ℒ(퐵, Ψ) + 퐶 (4.11)

for a constant 퐶 depending only on s and the choice of 퐵0.

If we write 퐴 = 퐴0 + 푎, then 4.8 becomes

(︂ )︂ + −1 * 1 + + SW(퐴, Φ) = 푑 푎 − 휌 ((ΦΦ ) ) + 퐹 푡 , 퐷 Φ + 휌(푎)Φ (4.12) 0 퐴 퐴0 2 0

We denote

^ (︁ + + )︁ 퐷(퐴, Φ) = 푑 푎, 퐷퐴0 Φ (︂ )︂ ^ 1 + −1 * 푄(퐴, Φ) = 퐹퐴푡 − 휌 ((ΦΦ )0), 휌(푎)Φ 2 0

We have

+ + −1 * * 퐷(퐴,Φ) SW(푎, 휑) = (푑 푎, 퐷퐴0 휑) + (−휌 (휑Φ + Φ휑 )0, 휌(푎)Φ + 휌(퐴 − 퐴0)휑) = 퐷^(푎, 휑) + 퐾^ (푎, 휑)

^ ^ where 퐾 = 퐷(퐴,Φ)푄; we denote this map by 퐷 SW. Denote also the restriction map on the tangent spaces by

2 1 + 2 1 푅 = 퐷ℛ : 퐿1(푋; 푖Λ ⊕ 푆 ) → 퐿1/2(푌 ; 푖Λ ⊕ 푆).

82 Proposition 4.9 ([Kha15, Proposition 3.1]). The map

퐷^ ⊕ Π−푅 : CoulCC(푋) → 퐿2(푋; 푖Λ+ ⊕ 푆− ⊕ 푖Λ0(푋)) ⊕ Coul(푌 ) (4.13) is Fredholm with index

+ + 2 indC(퐷퐴0 ) + 푏1(푋) − 푏 (푋) − 푏1(푌 ).

Moreover, ^ − ‖푥‖ 2 ≤ 퐶(‖퐷푥‖퐿2 + ‖Π 푅푥‖ 2 + ‖푥‖퐿2 ). 퐿1 퐿1/2

^ 2 1 + 2 + − 0 The map 퐾 : 퐿1(푋; 푖Λ ⊕ 푆 ) → 퐿 (푋; 푖Λ ⊕ 푆 ⊕ 푖Λ ) (and thus its restriction to the double Coulomb gauge) is compact by 12. Therefore

Corollary 4.10. The map

퐷 SW ⊕Π−푅 : CoulCC(푋) → 퐿2(푋; 푖Λ+ ⊕ 푆− ⊕ 푖Λ0) ⊕ Coul(푌 ) (4.14) is Fredholm with index as in Proposition 4.9.

Let us denote by 푋* = 푋 ∪[0, ∞)×푌 the manifold obtained from 푋 by attaching a cylindrical end with a product metric.

Proof of Theorem 6. Manifold. We need to show the surjectivity of

퐷 SW : CoulCC(푋) → 퐻 = 퐿2(푋; 푖Λ+ ⊕ 푆− ⊕ 푖Λ0).

Let us assume this map is not surjective. By Corollary 4.10 we know that the image of 퐷 SW is closed. Thus, there is 0 ̸= 푣 ∈ 퐻 orthogonal to im 퐷 SW, i.e. CC ^ ^ ⟨푣, 퐷 SW(푤)⟩퐿2 = 0 for any 푤 ∈ Coul (푋). Note that 퐷 SW(푤) = 퐷(푤) + 퐾(푤) ^ 2 1 + and 퐾 is a certain multiplication by 푝 = (퐴 − 퐴0, Φ) ∈ 퐿1(푋; 푖Λ ⊕ 푆 ). 2 * 2 * 1 + We first prove that 푣 ∈ 퐿1. Let us extend 푝 to 푝 ∈ 퐿1(푋 ; 푖Λ ⊕푆 ) in an arbitrary * 2 * + − 0 * way, and extend 푣 to 푣 ∈ 퐿 (푋 ; 푖Λ ⊕ 푆 ⊕ 푖Λ ) by putting 푣 |[0,∞)×푌 = 0. We still * ^ ^ * ^ * ^ * * have ⟨푣 , 퐷(푤)+퐾(푤)⟩퐿2(푋*) = 0 and therefore 푣 is a weak solution to (퐷 +퐾 )푣 =

83 ^ * ^ * ^ ^ ^ * 2 2 0 where 퐷 , 퐾 are formal adjoints of 퐷, 퐾, respectively. The map 퐾 : 퐿1 → 퐿 is * 2 * + compact by Theorem 12. Thus from Theorem 18 it follows that 푣 ∈ 퐿1(푋 ; 푖Λ ⊕ 푆− ⊕ 푖Λ0) and it is a solution to (퐷^ * + 퐾^ *)푣* = 0. Proposition A.12 implies that actually 푣* = 0 and therefore 푣 = 0. Thus, by contradiction, we have proved that 퐷 SW is surjective. What follows is that ℳ표(푋, s) is a Hilbert manifold. Axiom M1. Clearly, ℰ is bounded below on ℳ표(푋, s) (cf. Equation 4.10).

Therefore, by Equation 4.11, ℒ is bounded above on im ℛ(푋,s).

Suppose now that {ℛ(푋,s)(퐴푖, Φ푖)} converges weakly in im ℛ(푋,s). By Axiom F1, this implies that {ℒ(ℛ(푋,s)(퐴푖, Φ푖))} is bounded, and thus {ℰ(퐴푖, Ψ푖)} is bounded, too. The argument of [KM07, Theorem 5.1.1.ii.a] applied using the double Coulomb gauge instead of the Coulomb-Neumann gauge shows that after choosing a suitable

표 2 subsequence, (퐴푖, Φ푖) converges to (퐴, Φ) ∈ ℳ (푋, s) weakly in 퐿1 and strongly in 2 퐿 . Since, in any Hilbert space, weak convergence 푥푖 ⇀ 푥 implies lim inf ‖푥푖‖ ≥ ‖푥‖, we deduce from Equation 4.10 that lim inf ℰ(퐴푖, Φ푖) ≥ ℰ(퐴, Φ) which in turn implies 푤 lim sup ℒ(ℛ(푋,s)(퐴푖, Φ푖)) ≤ ℒ(lim ℛ(푋,s)(퐴푖, Φ푖)) = ℒ(ℛ(푋,s)(퐴, Φ)). Axiom M2. This is proven in the paragraph above. Axiom M3. Similarly to the argument above, after taking a subsequence in

표 im ℛ(푋,s), we can lift it to a weakly convergent sequence in ℳ (푋, s) using the argument of [KM07, Theorem 5.1.1.ii.a]; as in the proof of [KM07, Theorem 5.1.1.ii.b], we can further choose a strongly convergent subsequence.

퐷ℛ(푋,s) is a negative map. The Fredholmness follows directly from Corollary 4.10. Compactness also follows from the same argument, analyzing an Atiyah-Patodi- Singer boundary value problem as in [Kha15, Proposition 3.1] (c.f. also [KM07,

Proposition 17.2.5, Theorem 17.3.2]).

4.3 Moduli spaces are correspondences

In the previous section we discussed how a 4-manifold 푋 with boundary 푌 and spin푐 structure s gives rise to a cycle in B(푌, s|푌 ). If 푊 is a compact 4-manifold with two

84 푐 connected boundary components, 휕푊 = −푌1 ⊔ 푌2, and s is a spin structure on 푊 , we expect to obtain a correspondence from B(푌1, s|푌1 ) to B(푌2, s|푌2 ). This is indeed the case, but we need to refine the definition of the moduli space considered. Inthis section we also prove that composing two cobordisms gives a correspondence which is homotopic to the composition of the respective correspondences, thus showing the functoriality of the map induced on homology. We start by introducing the double Coulomb condition.

Definition 4.11 (double Coulomb slice).

{︂ ⃒ 표표 ⃒ * * ℬ (푊, s) = (퐴, Φ) ∈ 풞(푊, s)⃒푑 (퐴 − 퐴 ) = 0, 푑 (퐵 − 퐵 ) = 0, (4.15) ⃒ 푋 0 푌 0 ∫︁ }︂ (퐴 − 퐴0)(⃗푛) = 0 푌1 {︁ ⃒ }︁ ℎ 1⃒ * −1 풢 (푊 ) = 푢 : 푊 → 푆 ⃒푑푊 (푢 푑푢) = 0, 푢|휕푊 locally constant , (4.16)

The space ℬ표표(푊, s) of configurations in double Coulomb gauge on 푊 is being acted on by a continuous family of harmonic functions, the identity component 풢푒(푊 ) ∩ 풢ℎ(푊 ) of 풢ℎ(푊 ), where

푒 푓 2 풢 (푊 ) = {푒 : 푓 ∈ 퐿2(푊 ; 푖R)}.

Precisely, 푒푓 ∈ 풢ℎ(푊 ) whenever 푓 is a harmonic function which is locally constant on 휕푊 . Such function is determined by two values, 푓|푌1 ∈ 푖R and 푓|푌2 ∈ 푖R. For a manifold 푋 with single boundary component 푌 = 휕푋 this just corresponded to

1 the action of 푆 by multiplication on the spinor part, and thus fixing 푢|휕푋 = 1 and quotienting by the relative gauge group 풢ℎ(푋, 휕푋) gave us a space with 푆1-action which commuted with the restriction to the boundary. To obtain a manifold we still would like to quotient by a discrete group, but using

ℎ ℎ 1 풢 (푊, 휕푊 ) = {푢 ∈ 풢 : 푢|휕푊 = 1} ≃ 퐻 (푊, 휕푊 )

85 is not convenient as there seems to be no convenient gauge slice that would allow us to reduce to this action precisely because it contains the subgroup

→ 푓 2 * 풢 (푊 ) = {푒 : 푓 ∈ 퐿2(푊 ; 푖R), 푑 푑푓 = 0, 푓|푌1 = 0, 푓|푌2 = 2휋푖푘 ∈ 2휋푖Z} ≃ Z.

We would like to be left with an action of the quotient

ℎ → ℎ 푒 ℎ ℎ 1 풢 (푊, 휕푊 )/풢 (푊 ) ≃ 풢 (푊 )/(풢 (푊 ) ∩ 풢 (푊 )) = 휋0(풢 (푊 )) ≃ 퐻 (푊 ).

Notice that the following diagram commutes:

→ ˓→ ℎ ℎ 0 풢 (푊 ) 풢 (푊, 휕푊 ) 휋0(풢 (푊 )) 0

≃ ≃ ≃ 0 퐻˜ 0(휕푊 ) ˓→ 퐻1(푊, 휕푊 ) 퐻1(푊 ) 0.

A gauge group suitable to our situation can be obtained by choosing a section 푠 :

ℎ ℎ 1 휋0(풢 (푊 )) → 풢 (푊, 휕푊 ). This corresponds to choosing a section 푠˜ : 퐻 (푊 ) → 퐻1(푊, 휕푊 ). Since the Poincare pairing is perfect and is compatible with the maps ˜ 0 1 1 휕 ˜ 퐻 (휕푊 ) → 퐻 (푊, 휕푊 ) → 퐻 (푊 ) and 퐻1(푊 ) → 퐻1(푊, 휕푊 ) −→ 퐻0(휕푊 ), we 푓 푓 just need to choose 훼 ∈ 퐻1 (푊, 휕푊 ) (where 퐻* stands for the torsion-free part of homology) such that 휕훼 = 1 ∈ Z ≃ 퐻0(휕푊 ), where the last identification is given by a signed count of points on 푌2 ⊂ 휕푊 . Equivalently, we ask that ⟨[푑푓푊 ], 훼⟩ = 1 (if ∫︀ ∞ 훾 is a smooth curve representing 훼 then 푑푓푊 (훼) = 훾 푑푓푊 ) where 푓푊 ∈ 퐶 (푊, R) is * such that 푑 푑푓푊 = 0, 푓푊 |푌1 = 0 and 푓푊 |푌2 = 1, so that

풢→(푊 ) = ⟨푒2휋푖푓푊 ⟩.

1 1 Then 푠˜훼 : 퐻 (푊 ) → 퐻 (푊, 휕푊 ) is given by the condition that ⟨푠˜훼(푥), 훼⟩ = 0, and ℎ ℎ 푠훼 : 휋0(풢 (푊 )) → 풢 (푊, 휕푊 ) is given by the condition that

−1 ⟨[푢 푑푢], 훼⟩ = 0 for any 푢 = 푠훼(푥).

86 This condition is still not suitable for defining a gauge slice, so we proceed to rephrase it. Let us consider the space of doubly harmonic 1-forms

1 1 * ℋ (푊, 휕푊 ) = {휂 ∈ Ω (푊 ):Δ푊 휂 = 0, Δ휕푊 (휄휕푊 휂) = 0}

which by 푏1(푌푖) = 0 is equal to

1 1 * ℋ (푊, 휕푊 ) = {휂 ∈ Ω (푊 ):Δ푊 휂 = 0, 휄휕푊 휂 = 0},

the space of harmonic forms pulling back to 0 on the boundary. We have ℋ1(푊, 휕푊 ) ≃

퐻1(푊, 휕푊 ; R) and

1 ℎ 1 1 2휋푖퐻 (푊, 휕푊 ) ≃ 풢 (푊, 휕푊 ) ⊂ ℋ (푊, 휕푊 ) ≃ 퐻 (푊, 휕푊 ; R)

where the inclusion is given by

1 푢 ↦→ 푢−1푑푢 2휋푖

and indeed 풢ℎ(푊, 휕푊 ) is isomorphic to the lattice of forms 휂 ∈ ℋ1(푊, 휕푊 ) with

⟨휂, 훽⟩ ∈ Z for any 훽 ∈ 퐻1(푊, 휕푊 ). The condition

⟨휂, 훼⟩ = 0 cuts out a codimension-1 hypersurface in ℋ1(푊, 휕푊 ) which, since 훼 is an integral homology class, contains a corank-1 sublattice of 풢ℎ(푊, 휕푊 ), as shown before. How-

′ ∫︀ ′ ever, since the inner product ⟨휂, 휂 ⟩ = 푊 휂 ∧ *휂 is nondegenerate, there is a unique 1 휂훼 ∈ ℋ (푊, 휕푊 ) such that ⟨휂, 훼⟩ = ⟨휂, 휂훼⟩ and we get that

∫︁ ℎ ℎ −1 풢훼(푊, 휕푊 ) = {푢 ∈ 풢 (푊, 휕푊 ): 푢 푑푢 ∧ *휂훼 = 0}. 푊

We are now ready to define the appropriate gauge slice.

87 Definition 4.12 (moduli spaces on cobordisms). For any 푓 훼 ∈ 퐻1 (푊, 휕푊 ) such that ⟨푑푓푊 , 훼⟩ = 1 we define

{︂ ⃒ 표 ⃒ * * ℬ (푊, s) = (퐴, Φ) ∈ 풞(푊, s)⃒푑 (퐴 − 퐴 ) = 0, 푑 (퐵 − 퐵 ) = 0, (4.17) 훼 ⃒ 푋 0 푌 0 ∫︁ }︂ (퐴 − 퐴0) ∧ *휂훼 = 0 푊 표 표 ℳ̃︁훼(푊, s) = {(퐴, Φ) ∈ ℬ훼(푊, s)|SW(퐴, Φ) = 0} , (4.18) {︂ ⃒ ∫︁ }︂ ℎ 1⃒ * −1 −1 풢훼(푊, 휕푊 ) = 푢 : 푊 → 푆 ⃒푑푊 (푢 푑푢) = 0, 푢|휕푊 = 1, 푢 푑푢 ∧ *휂훼 = 0 , ⃒ 푊 (4.19)

표 표  ℎ ℳ훼(푊, s) = ℳ̃︁훼(푊, s) 풢훼(푊, 휕푊 ) . (4.20)

∫︀ −1 −1 Note that 푊 푢 푑푢 ∧ *휂훼 = ⟨[푢 푑푢], 훼⟩ and taking a smooth representative 훾 for

훼 = [훾] with 휕훾 = 푝2 − 푝1 (where 푝푖 ∈ 푌푖) we get that 푢(푝2) = 푢(푝1) and therefore

ℎ ∫︀ −1 Lemma 4.13. Assume 푢 ∈ 풢 (푊 ) satisfies 푊 푢 푑푢∧*휂훼 = 0. Then the conditions

푢|푌1 = 1, 푢|푌2 = 1 and 푢|휕푊 = 1 are pairwise equivalent.

Lemma 4.14. Any configuration in 풞(푊, s) can be transformed by some 푢 ∈ 풢(푊 )

표 푒 into ℬ훼(푊, s). Moreover, we can choose an assignment 퐴 ↦→ 푢훼(퐴) ∈ 풢 (푊 ) which is smooth.

Proof. The proof of Lemma 4.7 shows that the configuration can be put in the double Coulomb gauge by some 푢˜(퐴) ∈ 풢푒(푊 ) depending smoothly on 퐴. Then one can

∫︀ −1 휒(퐴)푓푊 take 휒(퐴) = 푊 (퐴 − 퐴0 − 푢˜ 푑푢˜) ∧ *휂훼 and 푢(퐴) = 푒 푢˜(퐴) is the desired gauge transformation since

∫︁ (−푑푓푊 ) ∧ *휂훼 = −⟨[푑푓푊 ], 훼⟩ = 1. 푊

88 Theorem 7. The restriction map

표 표 표 ℛ(푊,s,훼) : ℳ훼(푊, s) → ℬ (푌1, s|푌1 ) × ℬ (푌2, s|푌2 )

is an 푆1-equivariant dense correspondence of Floer spaces.

Proof. Repeating the proof of Theorem 6 almost verbatim (the additional gauge fixing condition is preserved under weak limits and only 1-dimensional, thus does not affect Fredholmness or compactness of the differentials) we obtain a stronger result, namely that

표 표 표 ℛ(푊,s,훼) : ℳ훼(푊, s) → ℬ (−푌1, s|푌1 ) × ℬ (푌2, s|푌2 )

∨ is a chain in (B(푌1, s|푌1 )) × B(푌2, s|푌2 ). The density of this correspondence is harder to achieve and we refer to Lipyanskiy

[Lip08, Theorem 9] for the proof.

푓 Proposition 4.15. Let 훼푖 ∈ 퐻1 (푊, 휕푊 ) be such that ⟨푑푓푊 , 훼푖⟩ = 1. Then the chains

ℛ(푊,s,훼1) and ℛ(푊,s,훼2) are homotopic.

̃︁표 Proof. For (퐴, Φ) ∈ ℳ훼1 (푊, s) we define

̃︁표 퐴훼1,훼2 (퐴, Φ) = 푢훼2 (퐴) · (퐴, Φ) ∈ ℳ훼2 (푊, s)

푓 using 푢훼2 constructed in Lemma 4.14. Precisely, since 훼2 − 훼1 ∈ 퐻1 (푊 ) and 퐴 − 퐴0 ′ 휒 (퐴)푓푊 is already in a double Coulomb gauge, we get 퐴훼1,훼2 (퐴, Φ) = 푒 (퐴, Φ) with

′ 휒 (퐴) = ⟨퐴 − 퐴0, 휂훼2 ⟩

= ⟨퐴 − 퐴0, 휂훼2 − 휂훼1 ⟩

= ⟨퐴 − 퐴0, 휂훼2−훼1 ⟩.

Note that 휒′(퐴) is a continuous linear function, in particular smooth.

89 We first check that this map is equivariant with respect to the actionof 퐻1(푊 ) ≃

ℎ ℎ ℎ 풢훼1 (푊, 휕푊 ) ≃ 풢훼2 (푊, 휕푊 ). This follows since, for 푢 ∈ 풢훼1 (푊, 휕푊 ) corresponding to 푥 ∈ 퐻1(푊 ) we have

′ −1 ′ −1 휒 (퐴 − 푢 푑푢) − 휒 (퐴) = ⟨푢 푑푢, 휂훼2 ⟩

−1 = ⟨푢 푑푢, 휂훼2 − 휂훼1 ⟩

= 2휋푖 ·⟨푥, 훼2 − 훼1⟩ ∈ 2휋푖 · Z, and thus

푢훼2 (푢 · 퐴) · 푢 · (퐴, Φ) =푢 ˜ · 푢훼2 (퐴) · (퐴, Φ) where

2휋푖⟨푥,훼2−훼1⟩·푓푊 ℎ 푢˜ = 푢 · 푒 ∈ 풢훼2 (푊, 휕푊 ) is in the same component of 풢(푊 ) as 푢 and thus corresponds to the same element 푥 ∈ 퐻1(푊 ). Now we check what this map does on the boundary. We have

휒′(퐴) ℛ(푊,s,훼2) ∘ 퐴훼1,훼2 (퐴, Φ) = 푒 ·ℛ(푊,s,훼1)(퐴, Φ)

푡휒′(퐴) and thus the family 푒 ·ℛ(푊,s,훼1)(퐴, Φ) gives a homotopy between the corresponding restriction maps.

We want to prove that the correspondence given by a composite cobordism 푊 =

푊1 ∪ 푊2, 휕푊푖 = −푌푖 ⊔ 푌푖+1, is equivalent to the fiber product of the two respective correspondences. Firstly, the exact sequence

푓 푓 0 = 퐻1 (푌 ) = 퐻2 (푊, 푊 ∖ 푌2) →

푓 →퐻1 (푊, 푌1 ∪ 푌3) → 푓 ˚ 푓 ˚ →퐻1 (푊, 푌1 ∪ 푊2 ∪ 푌3) ⊕ 퐻1 (푊, 푌1 ∪ 푊1 ∪ 푌3) → 푓 ˜ 푓 →퐻1 (푊, 푊 ∖ 푌2) = 퐻0 (푌2) = 0

90 ˚ together with the natural isomorphisms 퐻1(푊, 푌1 ∪ 푊2 ∪ 푌3) ≃ 퐻1(푊1, 휕푊1) and ˚ 푓 퐻1(푊, 푌1 ∪푊1 ∪푌3) ≃ 퐻1(푊2, 휕푊2) show that choosing 훼 ∈ 퐻1 (푊, 휕푊 ) is equivalent 푓 푓 to choosing a pair (훼1, 훼2) ∈ 퐻1 (푊1, 휕푊1) × 퐻1 (푊2, 휕푊2). We denote 훼푖 = 훼|푊푖 . Because of the gauge slice we took these may not be isomorphic. A solution in

표 ℳ̃︁훼(푊, s) in the chosen gauge, restricting to configurations 훾1, 훾3 on 푌1, 푌3, respectively, can be restricted to 푊 and then gauge transformed into ℳ̃︁표 (푊 , s ) without 푖 훼푖 푖 푖 changing the restrictions to 푌1 and 푌3. However, the restrictions of these configura- 1 푖푐 tions to 푌2 may differ by the action of 푆 , i.e., may be equal to 훾2 and 푒 훾2. We will instead show that composition of correspondences induced by 푊푖 is homotopic to the correspondence induced by 푊 .

Theorem 8. Let 푊 = 푊1 ∪푌2 푊2, where 푊푖 is a cobordism from 푌푖 to 푌푖+1.

Let s푖 = s|푊푖 and 훼푖 = 훼|푊푖 . Fix a smooth reference connection 퐴0 on 푊 and

use its pullbacks as reference connections 퐴0|푊1 , 퐴0|푊2 and 퐵1, 퐵2, 퐵2 on 푊1, 1 푊2 and 푌1, 푌2, 푌3, respectively. Then there is an 푆 -equivariant diffeomorphism 표 표 표 퐴 : ℳ (푊, s) → ℳ (푊 , s ) × 표 ℳ (푊 , s ) such that ℛ is 푊1,푊2 훼 훼1 1 1 ℬ (푌2,s|푌2 ) 훼2 2 2 (푊,s,훼) 1 (︁ )︁ 푆 -equivariantly homotopic to ℛ × 표 ℛ ∘ 퐹 . (푊1,s1,훼1) ℬ (푌2,s|푌2 ) (푊2,s2,훼2)

̃︁표 ̃︁표 ̃︁표 Proof. We will construct a map 퐴푊1,푊2 : ℳ훼(푊, s) → ℳ훼1 (푊1, s1) × ℳ훼2 (푊2, s2), 표 표 show that it is a smooth immersion onto ℳ (푊 , s )× 표 ℳ (푊 , s ), that it is 1 1 ℬ (푌2,s|푌2 ) 2 2 1 푆 -equivariant and that it intertwines the actions of 풢훼(푊, 휕푊 ) and 풢훼1 (푊1, 휕푊1) ×

풢훼2 (푊2, 휕푊2). Finally, we will construct the desired homotopy.

̃︁표 * 2 1 Let (퐴, Φ) ∈ ℳ훼(푊, s). Denote 푏(퐴, Φ) = 휄푌2 (퐴 − 퐴0) ∈ 퐿1/2(푌2, 푖Λ ).

Lemma 4.16. The 1-form 푏(퐴, Φ) is 퐶∞. Moreover, the map (퐴, Φ) ↦→ 푏(퐴, Φ) is

표 2 1 smooth as a map ℳ̃︁ (푊, s) → 퐿푠+1/2(푌2, 푖Λ ) for any 푠 ≥ 0.

Proof of the Lemma. Let 푊 ′ ⊂ 푊˚ be a closed submanifold. We will show that the

′ 표 2 ′ 1 + restriction to 푊 induces a smooth map ℳ̃︁ (푊, s) → 퐿푠+1(푊 ; 푖Λ ⊕ 푆 ). This, in

91 turn, factors as

표 표 2 ′ 1 + ℳ̃︁ (푊, s) → ℳ̃︁ 2 ′ (푊, s) → 퐿 (푊 ; 푖Λ ⊕ 푆 ). 퐿푠+1(푊 ) 푠+1

표 where ℳ̃︁ 2 ′ (푊, s) is the set of solutions to Seiberg-Witten equations on 푊 in 퐿푠+1(푊 ) 2 ′ double Coulomb gauge that restrict to 퐿푠+1 configurations on 푊 , understood as 2 a subset of the space of configurations on 푊 which restrict to 퐿푠+1 configurations on 푊 ′ with topology induced by both norms:

2 2 2 ‖(퐴, Φ)‖ 2 2 ′ = ‖(퐴 − 퐴0, Φ)‖ 2 + ‖(퐴 − 퐴0, Φ)‖ 2 ′ 퐿1(푊 )∩퐿푠+1(푊 ) 퐿1(푊 ) 퐿푠+1(푊 )

2 2 ′ which we shortly denote by 퐿1(푊 ) ∩ 퐿푠+1(푊 ). This is a complete Hilbert space 2 ′ and the restriciton map from it to 퐿푠+1(푊 ) is linear and continuous. Therefore 표 the problem is reduced to showing that ℳ̃︁ 2 ′ (푊, s) is a smooth submanifold 퐿푠+1(푊 ) 2 2 ′ 표 표 of 퐿1(푊 ) ∩ 퐿푠+1(푊 ) and that the identity map ℳ̃︁ (푊, s) → ℳ̃︁ 2 ′ (푊, s) is 퐿푠+1(푊 ) well-defined, continuous and smooth. Firstly, we have shown before that the differential of the Seiberg-Witten map is surjective as a map CoulCC(푊 ) → 퐿2(푊 ; 푖Λ+ ⊕ 푆− ⊕ 푖Λ0). The same proof goes through with minor adjustments to show that the same map is surjective when re-

CC CC ′ 2 + − 0 2 ′ + − stricted to Coul (푊 ) ∩ Coul (푊 ) → 퐿 (푊 ; 푖Λ ⊕ 푆 ⊕ 푖Λ ) ∩ 퐿푠(푊 ; 푖Λ ⊕ 푆 ⊕ 0 표 2 2 ′ 푖Λ ). Therefore ℳ̃︁ 2 ′ (푊, s) is a smooth submanifold of 퐿1(푊 ) ∩ 퐿푠+1(푊 ). 퐿푠+1(푊 ) 표 표 It remains to prove that the identity map ℳ̃︁ (푊, s) → ℳ̃︁ 2 ′ (푊, s) is well- 퐿푠+1(푊 ) defined, continuous and smooth. Firstly, for an elliptic equation on 푊 we have a strong regularity theorem which states that (c.f. [KM07]) for solutions to the

2 ′ 2 elliptic equation their 퐿푠+1(푊 )-norm is bounded by a multiple of 퐿1(푊 )-norm. Ap- plying this principle to the Seiberg-Witten equations with Coulomb gauge we get that

표 표 ℳ̃︁ (푊, s) → ℳ̃︁ 2 ′ (푊, s) is well-defined and continuous. Applying the same prin- 퐿푠+1(푊 ) 표 표 ciple to the tangent spaces at point 푝 we get that 푇푝ℳ̃︁ (푊, s) → 푇푝ℳ̃︁ 2 ′ (푊, s) 퐿푠+1(푊 ) is an isometry, thus a diffeomorphism.

Finally, in a neighborhood 푈 of 푝 we can parametrize 푈 ∩ℳ̃︁표(푊, s) by an open set

표 표 표 2 in 푇푝ℳ̃︁ (푊, s) utilizing the projection 푈 ∩ ℳ̃︁ (푊, s) → 푇푝ℳ̃︁ (푊, s) along 퐿 (푊 )-

92 표 orthogonal complement of 푇푝ℳ̃︁ (푊, s). This follows from applying the Implicit Func- tion Theorem. However, applying the Implicit Function Theorem when proving that

표 ℳ̃︁ 2 ′ (푊, s) is a manifold we can use the very same orthogonal complement. 퐿푠+1(푊 ) 표 표 Thus the identity map ℳ̃︁ (푊, s) → ℳ̃︁ 2 ′ (푊, s) near 푝 factors as 퐿푠+1(푊 )

표 ′ 표 푈 ∩ ℳ̃︁ (푊, s) → 푈 ∩ 푇푝ℳ̃︁ (푊, s)

id ′ 표 −→ 푉 ∩ 푇푝ℳ̃︁ 2 ′ (푊, s) 퐿푠+1(푊 ) 표 → 푉 ∩ ℳ̃︁ 2 ′ (푊, s) 퐿푠+1(푊 ) for some open neighborhoods 푈, 푈 ′, 푉, 푉 ′ of 푝, where we already showed the mid- dle identity map is smooth, and the two other maps are smooth since they are parametrizations coming from the Implicit Function Theorem. This finishes the proof of the Lemma.

∞ ∞ Take 푔 = 퐺푑Π푑(퐵 − 퐵0) ∈ 퐶 (푌2; 푖R). There exist unique 푓푖 ∈ 퐶 (푊1; 푖R) such that

푓1|푌1 = 0, 푓1|푌2 = 푔, Δ푓1 = 0 and similarly

푓2|푌2 = 푔, 푓2|푌1 = 0, Δ푓2 = 0.

2 2 Moreover, the map 푔 ↦→ 푓푖 is linear and continuous as a map 퐿푠+1/2 → 퐿푠+1 for 푠 ≥ 0.

We define 퐴푊1,푊2 via

표 ℳ̃︁훼(푊, s) ∋ (퐴, Φ) ↦→ (4.21)

푓 +휒 (퐴)푓 푓 +휒 (퐴)푓 +휒 (퐴) 표 표 1 1 푊1 2 2 푊2 1 ̃︁ ̃︁ ↦→(푒 (퐴|푊1 , Φ|푊1 ), 푒 (퐴|푊2 , Φ|푊2 )) ∈ ℳ훼1 (푊1, s1) × ℳ훼2 (푊2, s2),

where 휒1(퐴) = ⟨(퐴 − 퐴0)|푊1 − 푑푓1, 휂훼1 ⟩ and 휒2(퐴) = ⟨(퐴 − 퐴0)|푊2 − 푑푓2, 휂훼2 ⟩ depend smoothly on 퐴. Smoothness of this map follows from applying Lemma 4.16, Theorem 12 and Lemma A.4 to the subsequent steps. The image is contained in the fiber product since (푓1 + 휒1(퐴)푓푊1 )|푌2 = 푔 + 휒1(퐴) = (푓2 + 휒2(퐴)푓푊2 + 휒1(퐴))|푌2 . (︁ )︁ 휒 (퐴)+휒 (퐴) 표 1 2 Moreover, we get ℛ(푊1,s1,훼1) ×ℬ (푌2,s2) ℛ(푊2,s2,훼2) ∘ 퐹 (퐴, Φ) = (1, 푒 ) ·

93 푡(휒1(퐴)+휒2(퐴)) ℛ(푊,s,훼)(퐴, Φ) and the aforementioned homotopy is given by (퐴, Φ, 푡) ↦→ (1, 푒 )· 표 표 표 ℛ(푊,s,훼)(퐴, Φ) as a map from ℳ (푊, s) × 퐼 to ℬ (푌1, s) × ℬ (푌3, s).

We proceed to checking that this intertwines the actions of the gauge groups. Recall that we have canonical isomorphisms

ℎ 1 풢훼(푊, 휕푊 ) ≃ 퐻 (푊 )

and 풢ℎ (푊 , 휕푊 ) ≃ 퐻1(푊 ). 훼푖 푖 푖 푖

ℎ ℎ ℎ Therefore we can identify 풢훼(푊, 휕푊 ) with the product 풢훼1 (푊1, 휕푊1)×풢훼2 (푊2, 휕푊2) 1 1 1 via the isomorphism 퐻 (푊 ) ≃ 퐻 (푊1) × 퐻 (푊2) coming from the exact sequence

1 1 1 ≃ 1 2 0 = 퐻 (푌2) → 퐻 (푊1) × 퐻 (푊2) −→ 퐻 (푊 ) → 퐻 (푌2) = 0.

Note that identifying 퐻1(푀) = [푀, 푆1] we can interpret this isomorphism as mapping

1 1 1 ([푢], [푣]) ∈ [푊1, 푆 ] × [푊2, 푆 ] (where 푢, 푣 are chosen to be 1 on 푌2) to (푢.푣)] ∈ [푊, 푆 ]

where (푢.푣) denotes the map which restricts to 푢 on 푊1 and to 푣 on 푊2.

표 ℎ 1 Take any (퐴, Φ) ∈ ℳ̃︁훼(푊, s) and 푢 ∈ 풢훼(푊, 휕푊 ) corresponding to 푥 ∈ 퐻 (푊 ).

Equation (4.21) provides the formula for 퐴푊1,푊2 (퐴, Φ). Using the formula for (퐴, Φ)

훿푓1+(훿푐1)(푓푊1 −1) and 푢(퐴, Φ) we get that 퐴푊1,푊2 (푢(퐴, Φ)) =푢 ˜1퐴푊1,푊2 (퐴, Φ) where 푢˜1 = 푒 푢|푊1 with

* −1 (훿푓1)|푌1 = 0, (훿푓1)|푌2 = −퐺푑Π푑(휄푌2 (푢 푑푢)), Δ(훿푓1) = 0

and subsequently

−1 (훿휒1) = ⟨−(푢 푑푢)|푊1 − 푑(훿푓1), 휂훼1 ⟩.

ℎ Notice this 푢˜1 is independent of (퐴, Φ), as expected. We check that 푢˜1 ∈ 풢훼1 (푊1, 휕푊1).

Clearly from the construction, 푢˜1 is harmonic and restricts to 1 on 푌1. Furthermore,

94 on 푌2 we have

−1 −1 푢˜1 푑푌2 푢˜1 = 푑푌2 (훿푓1) + (훿휒1)푑푌2 푓푊1 + 푢 푑푌2 푢

−1 −1 = −푑퐺푑Π푑(푢 푑푌2 푢 + (훿휒1)푑푌2 (1) + 푢 푑푌2 푢 = 0

−1 −1 −1 because 푑퐺푑 = id|im Π푑 and Π푑(푢 푑푌2 푢) = 푢 푑푌2 푢 because 푑푌2 (푢 푑푌2 푢) = 0. Thus

푢˜1 is harmonic (constant) on 푌2. Moreover,

−1 −1 ⟨푢˜1 푑푢˜1, 휂훼1 ⟩ = ⟨푑푌2 (훿푓1) + (훿휒1)푑푌2 푓푊1 + 푢 푑푌2 푢, 휂훼1 ⟩

−1 = 훿휒1 + ⟨푑(훿푓1) + 푢 푑푢, 휂훼1 ⟩ = 0,

ℎ and Lemma 4.13 finishes the proof of 푢˜1 ∈ 풢훼1 (푊1, 휕푊1). We obtain 푢˜2 in the same manner, and since (훿푓1)|푌2 = (훿푓2)|푌2 we get that 푢˜2|푌2 =푢 ˜1|푌2 = 1 and using this ℎ we get 푢˜2 ∈ 풢훼(푊2, 휕푊2) as before. Finally, (˜푢1).(˜푢2) represents 푢 since the factors 훿푓 +(훿휒 )푓 훿푓 +(훿휒 )푓 +훿휒 (퐴) 푒 1 1 푊1 and 푒 2 2 푊2 1 can be homotoped to 1.

We show that this map is bijective onto the fiber product, following the argument in [Lip08], and we prove that the inverse map is continuous. Let (퐴푖, Φ푖) ∈

ℳ̃︁표 (푊 , s ) such that ℛ푡(퐴 , Φ ) = ℛ푠(퐴 , Φ ). These would give a configuration 훼푖 푖 푖 1 1 2 2 on 푊 if the normal components of connections 퐴1 and 퐴2 agreed on 푌2. Let ℎ1 푑푡 and ℎ2 푑푡 be the 푑푡-components of (퐴1 − 퐴0)|푌2 and (퐴2 − 퐴0)|푌2 . We want to find 2 harmonic functions 푓푖 ∈ 퐿2(푊푖; 푖R) such that

푓1|푌2 = 푓2|푌2 ,

휕푡푓1|푌2 + ℎ1 = 휕푡푓2|푌2 + ℎ2,

푓1|푌0 = 0, 푓2|푌2 = 0.

Take a tubular neighborhood [−휀, 휀] × 푌2 ⊂ 푊 of 푌2. Let {휑휆}휆 be an eigenbasis for

95 ∑︀ 2 ∑︀ 1/2 2 Δ푌2 and write ℎ2 − ℎ1 = 휆 푐휆휑휆. Since ℎ2 − ℎ1 ∈ 퐿1/2, therefore 휆 휆 |푐휆| < ∞ 2 and thus the following are well-defined as elements of 퐿2([−휀, 휀] × 푌2; 푖R):

1 1/2 푔 = ∑︁ 푐 휆−1/2푒휆 푡휙 , 1 2 휆 휆 1 1/2 푔 = ∑︁ 푐 휆−1/2푒−휆 푡휙 , 2 2 휆 휆

∞ which satisfies 휕푡푔1|푌2 − 휕푡푔2|푌2 = ℎ1 − ℎ0. Let 휌 ∈ 퐶 (푊 ; R) be a bump function supported in [−휀, 휀] × 푌2 which is identically 1 in a neighborhood of 푌2. The config-

휌푔1 휌푔2 ′ ′ urations 푒 (퐴1, Φ1) and 푒 (퐴2, Φ2) patch to give a configuration (퐴 , Φ ), but this ′ is not necessarily in Coulomb gauge becuase 휌푓푖 are not necessarily harmonic. Take 2 푓 ∈ 퐿2(푊 ; 푖R) such that

푓|푌1 = 0, 푓|푌3 = 0, Δ푓 = −Δ(휌푔1) − Δ(휌푔2)

and then 푐 = ⟨퐴1 − 퐴0, 휂훼|푊1 ⟩ + ⟨퐴2 − 퐴0, 휂훼|푊2 ⟩ + ⟨푑푓, 휂훼⟩. We get (퐴, Φ) =

푓+푐푓푊 ′ ′ 표 푒 (퐴 , Φ ) ∈ ℳ훼(푊, s) such that 퐴푊1,푊2 (퐴, Φ) = ((퐴1, Φ1), (퐴2, Φ2)).

2 2 Note that (푔1, 푔2) as elements of 퐿2 depend continuously on ℎ1 − ℎ0 ∈ 퐿1/2 which 2 2 in turn depends continuously on 퐴1 − 퐴0, 퐴2 − 퐴0 ∈ 퐿1/2. Moreover, 푓 ∈ 퐿3 depends 2 2 2 2 continuously on (푔1, 푔2) ∈ 퐿2. If the multiplication 퐿2 × 퐿1 → 퐿1 was continuous on four-manifolds then we would have shown that the map ((퐴1, Φ1), (퐴2, Φ2)) → (퐴, Φ)

푔푖 2 which we constructed is continuous. It remains to show that 푒 Φ푖 ∈ 퐿1 depends continuously on the initial configurations. We will prove it depends continuously on

2 2 ℎ2 − ℎ1 ∈ 퐿1/2 and Φ푖 ∈ 퐿1.

′ ′ ′ ′ Let (퐴1, Φ1) and (퐴2, Φ2) be another choice of configurations, and denote the

96 ′ ′ corresponding harmonic functions on [−휀, 0] × 푌2 and [0, 휀] × 푌2 by 푔1, 푔2. Then

′ ′ ′ 푔1 푔 ′ 푔 ′ 푔1 푔 ‖푒 Φ1 − 푒 1 Φ ‖ 2 ≤ ‖푒 1 (Φ1 − Φ )‖ 2 + ‖(푒 − 푒 1 )Φ1‖ 2 1 퐿1 1 퐿1 퐿1 푔′ 푔′ ′ ≤ 퐶(‖푒 1 ‖퐿∞ + ‖푒 1 ‖ 2 )‖Φ1 − Φ ‖ 2 퐿2 1 퐿1 ′ 푔1 푔 + 퐶‖푒 − 푒 1 ‖ 2 ‖Φ1‖ 2 퐿2 퐿1 ′ 푔1 푔 + 퐶‖푒 − 푒 1 ‖ ∞ ‖Φ1‖ 2 퐿 ([−휀,−훿]×푌2) 퐿1([−휀,−훿]×푌2) ′ 푔1 푔 + 퐶‖푒 − 푒 1 ‖ ∞ ‖Φ1‖ 2 퐿 ([−훿,0]×푌2) 퐿1([−훿,0]×푌2)

for any 훿. Now one can choose 훿 to have ‖Φ1‖ 2 as small as one wants while 퐿1([−훿,0]×푌2 푔 푔′ 푔 푔′ ′ 1 1 ∞ 1 1 ∞ ∞ ‖푒 −푒 ‖퐿 ([−훿,0]×푌2) ≤ 2. Moreover ‖푒 −푒 ‖퐿 ([−휀,−훿]×푌2) ≤ ‖푔1−푔1‖퐿 ([−휀,−훿]×푌2) ≤ ′ ′ 퐶‖(ℎ1 − ℎ2) − (ℎ − ℎ )‖ 2 via a direct computation (or by interior regularity esti- 1 2 퐿1/2 mates following from the Gårding inequality (Theorem 10).

Finally, we prove that the differential of this map, as a map onto the image,is

′ ′ 표 invertible. Take (퐴0 + 푎, Φ), (퐴0 + 푎 , Φ ) ∈ ℳ̃︁훼(푊, s). Denote 퐷 = ‖퐴푊1,푊2 (퐴0 + ′ ′ ′ ′ 푎, Φ)−퐴푊 ,푊 (퐴0+푎 , Φ )‖ 2 , assume 퐷 < 1 and and assume that ‖(푎−푎 , Φ−Φ )‖ 2 ≤ 1 2 퐿1 퐿1 퐷휀 for a constant 휀. We will later choose 휀 to obtain a contradiction, proving our

′ claim. We get that ‖푔 − 푔 ‖ 2 ≤ 퐶푘퐷휀 and thus 퐿푘

′ ′ ‖푓푖 − 푓 ‖ 2 ≤ 퐶‖푔 − 푔 ‖ 2 푖 퐿3 퐿5/2

≤ 퐶퐶5/2퐷휀.

Since 휒푖 are linear and continuous, we also have an estimate

′ ′ 휒푖(퐴0 + 푎) − 휒푖(퐴0 + 푎 ) ≤ 퐶 퐷휀

and by Sobolev multiplication theorem it follows that ‖퐴푊1,푊2 (퐴0+푎, Φ)−퐴푊1,푊2 (퐴0+ ′ ′ ′′ ′′ 1 푎 , Φ )‖ 2 ≤ 퐶 퐷휀 for some 퐶 depending on ‖Φ‖ 2 . Choosing 휀 = ′′ gives the 퐿1 퐿1 1+퐶 desired contradiction.

97 4.4 Existence of admissible perturbations

To be able to perturb maps for transversality while perserving the semi-infinite topology (preserving semi-compactness and polarization of maps while perturbing them) we need to find a family of admissible perturbations as described by Definition 3.17. We will prove the existence of admissible perturbations for the Seiberg-Witten-Floer spaces from which it follows that any correspondences going from them induce maps on homology and that there exists a Poincaré pairing between the homology of the Seiberg-Witten-Floer space and its dual. Let

2 1 * P = {(푏, 휓) ∈ 퐿 (푌 ; 푖Λ ⊕ 푆): 푑 푏 = 0, ‖(푏, 휓)‖ 2 < 1} (4.22) 3 퐿3

2 be the 퐿3-ball in the tangent space to B. Define Θ: B × P → B by

Θ(푥, 푝) = 푥 + 푝 (4.23)

2 2 with a slight abuse of notation, since we first map 푝 ∈ 퐿3 to itself in 퐿1/2 and then add it to 푥 ∈ ℬ표.

Theorem 9 (perturbations for Seiberg-Witten-Floer spaces). This Θ is a family of admissible perturbations for the Seiberg-Witten-Floer space B(푌, s).

Proof. Since Θ is bilinear and continuous, it is also smooth. Moreover, Θ(푥, 0) = 푥 by definition.

Axiom (P1) is satisfied since 푥푖 = Θ(푥푖, 푝푖) − 푝푖.

Axiom (P4) is satisfied since 퐷BΘ = id푇 푋 . 2 2 Axiom (P5) is satisfied since the inclusion 퐿3 → 퐿1/2 is compact by the Rellich lemma, Theorem 11.

2 2 Axiom (P6) is satisfied since the inclusion 퐿3 → 퐿1/2 is dense. We now show that Θ satisfies Axioms (P2) and (P3). Let us compute ℒ(Θ(푥, 푝))−

98 ℒ(푥), where 푥 = (퐵, Ψ) = (퐵0 + 푏, Ψ) and 푝 = (휂, 휓). Using the facts that 퐷퐵 is self-adjoint, 휌(푎) is anti-self-adjoint, and 푑*휂 = 0:

1 ℒ(Θ(푥, 푝)) − ℒ(푥) = (⟨퐷 (Ψ + 휓), Ψ + 휓⟩ − ⟨퐷 Ψ, Ψ⟩) 2 퐵0+푏+휂 퐵0+푏 1 (︂∫︁ ∫︁ )︂ − 2(푏 + 휂) ∧ (2퐹퐵푡 + 2푑(푏 + 휂)) − 2푏 ∧ (2퐹퐵푡 + 2푑푏) 8 푌 0 푌 0 1 (︂ = ⟨퐷 휓, 2Ψ + 휓⟩ + ⟨휌(푏)휓, 휓⟩ + ⟨휌(휂)(Ψ + 휓), Ψ + 휓⟩ 2 퐵0 ∫︁ ∫︁ )︂ +2 푏 ∧ 푑휂 + 휂 ∧ (퐹퐵푡 + 휂) 푌 푌 0

2 1 Since 퐿3(푌 ) ˓→ 퐶 (푌 ) is continuous (cf. Theorem 15) and the multiplication map 퐶0(푌 )×퐿2(푌 ) → 퐿2(푌 ) is continuous, therefore ℒ(Θ(푥, 푝))−ℒ(푥) is continuous with

2 2 2 2 respect to the 퐿 -norm in 푥 and 퐿3-norm in 푝. But 퐿1/2(푌 ) → 퐿 (푌 ) is compact and therefore ℒ(Θ(푥, 푝)) − ℒ(푥) is continuous with respect to the product of weak topology on 푋 and strong topology on P, that is, Axiom (P3) is satisfied. From the computation above we also deduce that

|ℒ(Θ(푥, 푝)) − ℒ(푥)| ≤‖푝‖ 2 · (‖푥‖ 2 + ‖푝‖ 2 ) + ‖푝‖ 0 ·‖푥‖ 2 ·‖푝‖ 2 퐿1 퐿 퐿 퐶 퐿 퐿 2 + ‖푝‖ 0 · (‖푥‖ 2 + ‖푝‖ 2 ) + 2‖푥‖ 2 ‖푝‖ 2 + ‖푝‖ 2 (‖퐹 푡 ‖ 2 + ‖푝‖ 2 ) 퐶 퐿 퐿 퐿 퐿1 퐿1 퐵0 퐿 퐿1 2 ≤퐶(‖푥‖퐿2 + 1)

2 ≤퐶(‖푥‖ 2 + 1) 퐿1

for some 퐶, since ‖푝‖ 2 < 1 and thus ‖푝‖ 0 , ‖푝‖ 2 , ‖푝‖ 2 are uniformly bounded. 퐿3 퐶 퐿1 퐿 Axiom (P2) follows.

4.5 Invariance under perturbations of the functional

In [KM07, Section 11], a family of perturbations used to define Monopole Floer Ho- mology is constructed. In this section we show that these perturbations do not change the semiinfinite homology.

99 Kronheimer and Mrowka first define so-called cylinder functions ([KM07, Defini- tion 11.1.1]) such that that the gradient of each cylinder function is a tame perturbation ([KM07, Proposition 11.1.2]). To achieve transversality, they provide a specific

countable family {푓푖} of such cylinder functions (cf. [KM07, Definition 11.6.3]) and use their gradients q푖 = grad퐿2 푓푖 to span a Banach space of perturbations utilizing ∑︀ the construction in [KM07, Theorem 11.6.1]. The space consists of all series 휆푖q푖 ∑︀ such that the norm 퐶푖|휆푖| is finite, for a suitably chosen sequence {퐶푖}. Note that ∑︀ assuming that the numbers 퐶푖 are large enough each such perturbation q = 휆푖q푖 is 2 ∑︀ an 퐿 -gradient of a function 푓 : 풞(푌, s) → R, 푓 = 휆푖푓푖.

Proposition 4.17. The numbers 퐶푖 can be chosen to be large enough so that every ∑︀ 표 ∑︀ 푓 = 휆푖푓푖 : ℬ (푌, s) → R satisfying 퐶푖|휆푖| < ∞ is bounded and weakly continuous.

∑︀ Proof. We can choose 퐶푖 large enough so that the series 휆푖푓푖 is uniformly convergent. Therefore it suffices to prove that each cylinder function 푓 is bounded and weakly continuous.

Each cylinder function is defined as 푓 = 푔 ∘ 푝, where 푝 : 풞(푌, s) → R푛 × T × C푚 with T = 퐻1(푌, 푖R)/(2휋푖퐻1(푌, Z)), and 푔 : R푛 × T × C푚 → R being an 푆1-invariant, smooth, compactly supported function. Since 푔 is continuous and compactly supported, it is bounded, and thus 푓 is bounded. It remains to prove weak continuity of 푝 in order to show weak continuity of 푓.

The first 푛 coordinates of 푝 are obtained by choosing 1-forms 푐1, . . . , 푐푛 and taking the 2 퐿 product, (퐵, Ψ) ↦→ ⟨푐푖, 퐵 − 퐵0⟩퐿2(푌 ), so this component is weakly continuous. For 표 푏1 = 0 the component T = {*} is trivial. Finally, on the Coulomb slice ℬ (푌, s) the last 푛 coordinates are obtained by choosing sections ϒ1,..., ϒ푚 of the spinor bundle 2 and taking the 퐿 product again, (퐵, Ψ) ↦→ ⟨ϒ푖, Ψ⟩퐿2(푌 ), so this component is weakly continuous as well.

100 Appendix A

Appendix

Theorem 10 (Gårding inequality). Let 퐷 be a first-order elliptic operator with smooth coefficients on a manifold 푀 and 푀 ′ ⊂ 푀 be open with compact 푝 closure. Then there is a constant 퐶 such that for any 훾 ∈ 퐿푘+1,푙표푐(푋) we have

‖훾‖ 푝 ′ ≤ 퐶(‖퐷훾‖ 푝 + ‖훾‖퐿푝(푀)). 퐿푘+1(푀 ) 퐿푘(푀)

A.1 Sobolev spaces

For the following, assume 푀 is a smooth Riemannian manifold of dimension 푛 with cylindrical ends, possibly with boundary.

Theorem 11 (Sobolev embedding theorem/Rellich lemma). There is 푝 푞 a continuous inclusion 퐿푘(푀) ˓→ 퐿푙 (푀) provided that 푘 ≥ 푙, 푝 ≤ 푞 and either 푘 − 푛/푝 > 푙 − 푛/푞 or 푘 − 푛/푝 = 푙 − 푛/푞 with 1 < 푝 ≤ 푞 < ∞. If 푀 is compact, same holds without requiring 푝 ≤ 푞. Moreover, if 푘 − 푛/푝 > 푙 − 푛/푞, then the map is compact.

101 Proof. See [Pal68, Theorem 9.1] and [KM07, Theorem 13.2.1].

Theorem 12 (Sobolev multiplication theorem). Assume 푘, 푙 ≥ 푚 and 1/푝 + 1/푞 ≥ 1/푟 for 푝, 푞, 푟 ∈ (1, ∞). Then the multiplication

푝 푞 푟 퐿푘(푀) × 퐿푙 (푀) → 퐿푚(푀)

is continuous if any of these hold:

(a) (푘 − 푛/푝) + (푙 − 푛/푞) ≥ 푚 − 푛/푟 and both 푘 − 푛/푝 < 0, 푙 − 푛/푞 < 0,

(b) min(푘 − 푛/푝, 푙 − 푛/푞) ≥ 푚 − 푛/푟 and either 푘 − 푛/푝 > 0 or 푙 − 푛/푞 > 0,

What is more, whenever it is continuous, it restricts to a compact map on {푓} ×

푞 푟 퐿푙 (푀) → 퐿푚(푀) provided that 푙 > 푚 and 푙 − 푛/푞 > 푚 − 푛/푟.

Proof. See [Pal68] and [KM07, Theorem 13.2.2] for proofs.

Theorem 13 (product estimates). Let 1 < 푝, 푞1, 푞2 < ∞, 1 < 푟1, 푟2 ≤ ∞, 1 = 1 + 1 = 1 + 1 , 푠 > 0, 푀 = 푛. Then there exists 퐶 such that 푝 푞1 푟1 푞2 푟2 R

푠 푠 푠 ‖|∇| (푓푔)‖퐿푝(푀) ≤ 퐶(‖|∇| 푓‖퐿푞1 (푀)‖푔‖퐿푟1 (푀) + ‖푓‖퐿푞2 (푀)‖|∇| 푔‖퐿푟2 (푀)).

102 Proposition A.1. The multiplication

2 ∞ 2 2 (퐿 푛 (푀) ∩ 퐿 (푀)) × 퐿 푛−2 (푀) → 퐿 푛−2 (푀) 2 2 2

2 ∞ is well-defined and continuous with respect to the sumof 퐿 푛 and 퐿 norms on the 2 first factor.

2푛 2 푛 2 푛−2 Proof. Firstly notice that 퐿 푛−2 (푀) ˓→ 퐿 (푀) and 퐿1(푀) ˓→ 퐿 (푀) via Theorem 2

11. Then the Proposition follows from Theorem 13 by taking 푝 = 2, 푞1 = 2, 푟1 = ∞, 2푛 2 2 ∞ 푞2 = 푛, 푟2 = 푛−2 for 푓 ∈ 퐿 푛−2 (푀) and 푔 ∈ 퐿 푛 (푀) ∩ 퐿 (푀). 2 2

Theorem 14 (Sobolev trace theorem). 푁 ⊂ 푀 closed 푛 − 푗-dimensional smooth submanifold, 1 ≤ 푝 < ∞, and 푘, 푙 nonnegative integers with 푘 − 푗/푝 ≥ 푙 > 0. Then the restriction map extends to continuous

푝 푝 퐿푘(푀) → 퐿푙 (푁).

Proof. See [Pal68, Theorem 9.3].

Theorem 15 (Sobolev embedding/Morrey inequality). Assume 푠 ≥ 0, 1 ≤ 푝 < ∞, 푘 nonnegative integer and 훼 ∈ (0, 1). Suppose 푠 − 푛/푝 ≥ 푘 + 훼. Then we have a continuous embedding

푝 푘,훼 퐿푠(푀) ⊂ 퐶 (푀).

푝 푘 In particular, if 푠 − 푛/푝 > 푘, then 퐿푠(푀) ⊂ 퐶 is continuous (and compact if 푀 is compact).

Proof. See [Pal68, Theorem 9.2].

103 A.2 Infinite-dimensional manifolds

Theorem 16 (Smale [Sma65]). Let 푀, 푌 be Hilbert manifolds. Let 푓 : 푀 → 푌 be a 퐶푞 Fredholm map with 푞 > max(ind 푓, 0). Then the set of regular values of 푓 is a residual subset of 푌 .

From the above, we will deduce the following vesion of the transversality theorem.

Theorem 17. Let 푀, 푌 be Hilbert manifolds and 푍 ⊂ 푌 a Hilbert submanifold. Let 푓 : 푀 × 푃 → 푌 be a 퐶∞ map transversal to 푍 ⊂ 푌 such that the projection 휋 : 푓 −1(푍) → 푃 is Fredholm. Then for a residual subset of 푝 ∈ 푃 , the map

푓푝 = 푓(·, 푝): 푀 → 푌 is transversal to 푍.

Proof. Due to the Smale-Sard theorem 16, there is a residual set 푈 ⊂ 푃 such that

−1 −1 for each 푝 ∈ 푈 and 푥 ∈ 푓 (푍) with 휋(푥) = 푧, the map (퐷휋)푥 : 푇푥(푓 (푍)) → 푇푝푃 is surjective. We want to prove that for any 푝 ∈ 푈 and any 푥 ∈ 푓 −1(푍) such that 휋(푥) = 푝,

the map (퐷푓푝)푥 : 푇푥푀 → 푁푦푍 = 푇푦푌/푇푦푍 is onto, where 푦 = 푓(푥). Indeed, take

any [푣] ∈ 푁푦푍. We know there is 푤 ∈ 푇푥(푀 × 푃 ) such that [(퐷푓)푥(푤)] = [푣]. Take −1 푧 ∈ 푇푥푓 (푍) such that 퐷휋(푧) = 퐷휋(푤), which is possible by the previous paragraph. We have [퐷푓(푤 − 푧)] = [퐷푓(푤)] = [푣] in 푁푍 since 퐷푓(푧) ∈ 푇 푍. On the other hand,

푤 − 푧 ∈ 푇푥푀 since 퐷휋(푤 − 푧) = 0. This finishes the proof.

푘 Definition A.2퐶 ( 푙표푐 topology). Let 푀, 푁 be Hilbert manifolds and consider a se- 푘 푘 quence 푓푖 ∈ 퐶 (푀, 푁). We say that the sequence 푓푖 converges to 푓푖0 in 퐶푙표푐 if for each point 푚 ∈ 푀 there exists an open neighborhood 푚 ∈ 푈 ⊂ 푀 such that for

every sequence 푖 → 푖0, we have ‖푓푖 − 푓푖0 ‖퐶푘(푈,푁) → 0.

104 Proposition A.3. Let 푀, 푁 be Hilbert manifolds and 퐿 be a finite-dimensional man-

푘+1 푘 ifold. Assume 푓 ∈ 퐶 (푀 × 퐿, 푁). Then 푓 ∈ 퐶(퐿, 퐶푙표푐(푀, 푁)).

Proof. It suffices to prove that for each (푚0, 푙0) ∈ 푀×퐿 there is an open neighborhood 푘 (푚0, 푙0) ∈ 푈 × 푉 ⊂ 푀 × 퐿 such that 푓 ∈ 퐶(푉, 퐶 (푈, 푁)). This is a local problem, so by taking 푈 × 푉 small enough and appropriate charts, we can assume 푀, 푁 are

푛 open balls in a Hilbert space and 퐿 = 퐵(푙0, 푅) is an open ball in R for some 푛, with radius 푅. Since 푓 ∈ 퐶푘+1(푀 × 퐿, 푁), by taking (푈 × 푉 ) small enough we can assume that

푎 there exists 0 < 퐶 < ∞ such that sup푈×푉 ‖퐷 푓‖ < 퐶 for any 0 ≤ 푎 ≤ 푘 + 1. Define

푙−푙0 푓푙(푚) = 푓(푚, 푙). For 푙 ̸= 푙0, define [푙0, 푙] = {푡푙0 + (1 − 푡)푙 : 푡 ∈ [0, 1]} and 푣푙 ,푙 = . 0 |푙−푙0| Let 0 ≤ 푎 ≤ 푘. Then

∫︁ 푎 푎 푎 sup ‖퐷푀 푓푙(푚) − 퐷푀 푓푙0 (푚)‖ = sup ‖ 휕푣퐷푀 푓휆(푚) 푑휆‖ 푚∈푈 푚∈푈 [푙0,푙] 푎+1 ≤ sup |푙 − 푙0| · ‖퐷 푓(푚, 휆)‖ (푚,휆)∈푈×푉

≤ |푙 − 푙0| · 퐶

푘 which proves that 푓푙 converge to 푓푙0 in 퐶 (푈, 푁).

∏︀푛 Lemma A.4. Let 퐵, 퐵1, . . . , 퐵푛 be Banach spaces and 퐿 : 푖=1 퐵푖 → 퐵 be a continuous multilinear map. Then 퐿 is smooth as a map of Banach manifolds.

Proof. The derivative of 퐿 exists and at point 푏1, . . . , 푏푛 is given by (푣1, . . . , 푣푛) ↦→

퐿(푣1, 푏2, . . . , 푏푛) + 퐿(푏1, 푣2, 푏3, . . . , 푏푛) + ... + 퐿(푏1, . . . , 푏푛−1, 푣푛), so 퐿 is continuously

differentiable. This is a finite sum of maps that are continuous and linearin (푏1, . . . , 푏푛,

푣1, . . . , 푣푛), so each one has a derivative in (푏1, . . . , 푏푛) which is still linear in (푣1, . . . , 푣푛), so 퐿 is twice continuously differentiable. Iterating this procedure gives the result.

Corollary A.5. The Sobolev embedding, multiplication and trace maps are smooth whenever they are continuous. For any codimension-0 submanifold 푀 ′ ⊂ 푀, possibly with boundary, the restric-

푝 푝 ′ tion maps 퐿푠(푀) → 퐿푠(푀 ) are smooth.

105 A.3 Weakly convergent operators

This section follows [Lip08].

Definition A.6. [weak convergence of operators] A sequence of operators 퐴푖 : 푉 → 푊 between Hilbert spaces 푉, 푊 is said to converge weakly if there exists a bounded

operator 퐴 : 푉 → 푊 such that for each 푣 ∈ 푉 one has 퐴푖(푣) → 퐴(푣). We denote it

퐴푖 ⇀ 퐴.

Lemma A.7. Suppose the sequence of Hilbert space operators {퐴푖 : 푉 → 푊 } is

uniformly bounded. If 퐴푖 ⇀ 퐴 and 퐾 : 푊 → 푈 is compact, then 퐴푖 ∘ 퐾 → 퐴 ∘ 퐾.

Proof. Without loss of generality, we can assume 퐴 = 0. Suppose, on the contrary,

that there is a sequence {푣푖} with ‖푣푖‖ = 1 and ‖퐴푖 ∘ 퐾(푣푖)‖ ≥ 퐶 > 0 for each 푖. By

taking a subsequence we can assume that 퐾(푣푖) → 푤. We get

lim 퐴푖 ∘ 퐾(푣푖) lim 퐴푖(푤푖) = lim 퐴푖(푤) + lim 퐴푖(푤푖 − 푤) = 0

since 퐴푖 ⇀ 퐴 implies lim 퐴푖(푤) = 0 and on the other hand from uniform boundedness of {퐴푖} we obtain ‖퐴푖(푤푖 − 푤)‖ ≤ 퐷‖푤푖 − 푤‖ → 0.

Lemma A.8. Suppose the sequence of Hilbert space operators {퐴푖 : 푉 → 푊 } is * * uniformly bounded. If 퐴푖 ⇀ 퐴, 퐴푖 ⇀ 퐴 and 퐾 : 푊 → 푈 is compact, then 퐾 ∘ 퐴푖 → 퐾 ∘ 퐴.

* * * * Proof. From A.8 we deduce that 퐴푖 ∘ 퐾 → 퐴푖 ∘ 퐾 and the conclusion follows.

A.4 Regularity of solutions

Let 푋 be a smooth Riemannian manifold with asymptotically cylindrical ends (with-

∞ ∞ out boundary). Let 퐷0 be an elliptic operator 퐷0 : 퐶 (푋; 퐸) → 퐶 (푋; 퐹 ) of order 푚 which is asymptotically cylindrical on the ends of 푋.

106 푚푖푛 푚푎푥 2 Proposition A.9. The minimal and maximal operators 퐷0 , 퐷0 of 퐷0 : 퐿 (푋; 퐸) → 2 2 퐿 (푋; 퐹 ) coincide, and their domains are equal to 퐿푚(푋; 퐸).

Proof. This follows from [Shu92, Section 1, Proposition 4.1] because 푋 has bounded geometry and 퐷0 is uniformly elliptic.

Corollary A.10. Suppose in addition that 퐸 = 퐹 , that 퐸 is equipped with asymptotically cylindrical metric, and that 퐷0 is formally self-adjoint with respect to the 2 induced scalar product on 퐿 (푋; 퐸). Then 퐷0 is essentially self-adjoint.

Proposition A.11. For every 푠, 푡 ∈ R there exists 퐶 > 0 such that for any 푢 ∈ ∞ 퐶0 (푋; 퐸)

‖푢‖ 2 ≤ 퐶(‖퐷0푢‖ 2 + ‖푢‖ 2 ). 퐿푚+푠 퐿푠 퐿푡

Proof. This follows from [Shu92, Appendix 1, Lemma 1.4] since 퐷0 is uniformly elliptic.

2 2 Let 푚 = 1 and let 퐾 : 퐿1(푋; 퐸) → 퐿 (푋; 퐹 ) be a compact operator which has * 2 2 a compact formal adjoint 퐾 : 퐿1(푋; 퐸) → 퐿 (푋; 퐹 ). Consider the operator

2 2 퐷 = 퐷0 + 퐾 : 퐿1(푋; 퐸) → 퐿 (푋; 퐹 ).

2 2 Theorem 18. Assume that 푣 ∈ 퐿푙표푐(푋; 퐸) satisfies 퐷푣 = ℎ for ℎ ∈ 퐿 (푋; 퐹 ). 2 Then 푣 ∈ 퐿1(푋; 퐸). 2 2 Moreover, if 푣 ∈ 퐿푙표푐(푋; 퐸), then 푣 ∈ 퐿1,푙표푐(푋; 퐸).

Proof. By considering

⎛ ⎞ ⎛ ⎞ 0 퐷* 0 퐾* ⎜ 0⎟ ⎜ ⎟ 2 2 ⎝ ⎠ + ⎝ ⎠ : 퐿1(푋; 퐸 ⊕ 퐹 ) → 퐿 (푋; 퐸 ⊕ 퐹 ) 퐷0 0 퐾 0

107 * where 퐷0 is the formal adjoint of 퐷0 we may assume, without loss of generality, that

퐸 = 퐹 and 퐷0 is formally self-adjoint and thus, by Corollary A.10, is self-adjoint as 2 2 a map 퐿1(푋; 퐸) → 퐿 (푋; 퐸).

Consider the equation

(퐷0 + 푖푐)휓 + 퐾휓 = ℎ + 푖푐푣 (A.1) for large 푐 ∈ R. Certainly, 휓 = 푣 ∈ 퐿2(푋; 퐸) solves the equation. We claim that for 2 large 푐 this has a unique solution 휓 = 푢 ∈ 퐿1(푋; 퐸).

2 2 Consider the operator 퐷0 + 푖푐 : 퐿1(푋; 퐸) → 퐿 (푋; 퐸). Since the spectrum of −1 2 2 1 퐷0 is real, (퐷0 + 푖푐) : 퐿 (푋; 퐸) → 퐿 (푋; 퐸) exists and is bounded by |푐| (cf. [Lan93, Chapter XIX, Theorem 2.4]).

Moreover, we have the inequality ‖푢‖ 2 ≤ 퐶(‖퐷푢‖ 2 + ‖푢‖ 2 ) for any 푢 ∈ 퐿1 퐿 퐿 2 퐿1(푋; 퐸). Therefore

−1 −1 −1 ‖(퐷0 + 푖푐) 푢‖ 2 ≤ ‖퐷0(퐷0 + 푖푐) 푢‖ 2 + ‖(퐷0 + 푖푐) 푢‖ 2 퐿1 퐿 퐿 −1 ≤ ‖푢 − 푖푐(퐷0 + 푖푐) 푢‖퐿2 + (1/|푐|)‖푢‖퐿2

≤ (2 + 1/|푐|)‖푢‖퐿2

−1 2 2 and therefore (퐷0 + 푖푐) is a bounded operator 퐿 (푋; 퐸) → 퐿1(푋; 퐸).

Finally, by [Lan93, Chapter XIX, Theorem 2.7] we have an orthogonal decom- ^؀ 2 position 퐿 (푋; 퐸) = 퐻푛 such that the restriction 퐷푛 = 퐷0|퐻푛 : 퐻푛 → 퐻푛 is ؀ 2 a bounded operator (in 퐿 -norm on 퐻푛) and 퐷0 = ^ 퐷푛. Note that this implies that 2 퐻푛 ⊂ Dom(퐷0) = 퐿1(푋; 퐸). For each 푣푛 ∈ 퐻푛 we thus have, by Proposition A.11,

′ ‖푣푛‖ 2 ≤ 퐶(‖퐷0푣푛‖ 2 + ‖푣푛‖ 2 ) ≤ 퐶(퐶푛 + 1)‖푣푛‖ 2 = 퐶 ‖푣푛‖ 2 , 퐿1 퐿 퐿 퐿 푛 퐿

where 퐶푛 = ‖퐷푛‖퐿2 . We get that

′ −1 ′ −1 퐶푛 |푐|→∞ ‖(퐷0 + 푖푐) 푣푛‖퐿2 ≤ 퐶 ‖(퐷0 + 푖푐) 푣푛‖퐿2 ≤ ‖푣푛‖퐿2 −−−−→ 0 (A.2) 1 푛 |푐|

108 2 −1 |푐|→∞ 2 which proves that for any 푣 ∈ 퐿 (푋; 퐸), (퐷0 + 푖푐) 푣 −−−−→ 0 in 퐿1(푋; 퐸) and −1 therefore (퐷0 + 푖푐) converges weakly to 0 as |푐| → ∞. −1 2 2 By Lemma A.7 we conclude that 푇 = (퐷0 + 푖푐) 퐾 : 퐿1(푋; 퐸) → 퐿1(푋; 퐸) −1 2 converges strongly to 0. Thus, for large |푐|, the operator 퐼+(퐷0+푖푐) 퐾 : 퐿1(푋; 퐸) → 2 2 2 퐿1(푋; 퐸) is invertible, and therefore 퐷0 + 푖푐 + 퐾 = 퐷 + 푖푐 : 퐿1(푋; 퐸) → 퐿 (푋; 퐸) 2 is invertible, too. This establishes the existence of unique 휓 ∈ 퐿1(푋; 퐸) solving Equation (A.1). We now need to establish the uniqueness of solution to A.1 in 퐿2(푋; 퐸). We have

2 that (퐷0 + 푖푐 + 퐾)(휓 − 푣) = 0. This implies that 휓 − 푣 is perpendicular in 퐿 (푋; 퐸) * * 2 2 to the image of 퐷0 + 푖푐 + 퐾 = 퐷0 + 푖푐 + 퐾 : 퐿1(푋; 퐸) → 퐿 (푋; 퐸). However, we already established that the latter operator is invertible and it follows that 휓 −푣 = 0;

2 thus 푣 = 휓 ∈ 퐿1(푋; 퐸). 2 It remains to consider the case 푔 ∈ 퐿푙표푐. But then for any compact 퐴 ⊂ 푋 we can take a compactly supported bump function 휌 : 푋 → [0, 1] with 휌|퐴 = 1 and define 푣′ = 휌푣 ∈ 퐿2(푋; 퐸). We have then 퐷푣′ = ℎ′ with ℎ′ ∈ 퐿2(푋; 퐸) and therefore

′ 2 2 푣 ∈ 퐿1(푋; 퐸), so 푣 ∈ 퐿1(퐴; 퐸|퐴). Repeating the argument for every compact 퐴 ⊂ 푋 2 proves that 푣 ∈ 퐿1,푙표푐(푋; 퐸).

A.5 Unique continuation

Let 푢 : 퐼 = [푡1, 푡2] → 퐻 where 퐻 is a Hilbert space and 퐿(푡) be an unbounded linear operator in 퐻 with domain 퐷 (independent of 푡 ∈ 퐼). Assume that 푢(푡) ∈ 퐷 for any 푡 ∈ 퐼, 푢 ∈ 퐶1(퐼, 퐻) and 퐿(푡)푢(푡) ∈ 퐶(퐼, 퐻). Furthermore, assume that

1. For some Φ ∈ 퐿2(퐼),

‖휕푡푢 − 퐿(푡)푢(푡)‖ ≤ Φ(푡)‖푢(푡)‖

for any 푡 ∈ 퐼,

퐿(푡) = 퐿+(푡) + 퐿−(푡)

109 with 퐿+(푡) symmetric and 퐿−(푡) bounded and skew-adjoint,

3. the limit 퐿(푡 + ℎ)푥 − 퐿(푡)푥 퐿˙ (푡)푥 := lim ℎ→0 ℎ exists for all 푥 ∈ 퐷 and 푡 ∈ 퐼,

4. there exist constants 퐶1, 퐶2 such that

˙ ‖퐿(푡)푥‖ ≤ 퐶1‖퐿(푡)푥‖ + 퐶2‖푥‖

for any 푡 ∈ 퐼 and 푥 ∈ 퐷.

Proposition A.12. Let 푢 : 퐼 → 퐻 be a solution of the equation

푑푢 + 퐿(푡)푧 = Φ(푡) 푑푡

2 where 퐿(푡) satisfies the hypotheses above and Φ ∈ 퐿 (퐼, 퐻). If 푢(푡0) = 0 for any

푡0 ∈ 퐼, then 푢 is identically zero.

Proof. The proof of Lemma 7.1.3 in [KM07] goes through verbatim (with 푧 replaced by 푢) with the exception that the inequality log ‖푢(푡)‖ ≥ 푙(푡) − 훿|푡 − 푡0| is substituted

2 1 2 by log ‖푢(푡)‖ ≥ 푙(푡) − ‖Φ‖퐿 (퐼,퐻)|푡 − 푡0| since ‖Φ‖퐿 ([푡,푡0],퐻) ≤ ‖Φ‖퐿 ([푡,푡0],퐻)|푡 − 푡0|.

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