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Semi-Infinite Homology of Floer Spaces Piotr Suwara

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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-di- mensional 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 ho- mology 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 man- ifold 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 func- tional 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 di- mensional 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 pos- sibility of using such chains to define Floer homology [Ati88]. The idea for this con- struction 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 compacti- fying 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 transver- sality 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.
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