ASSEMBLY MAPS AND PSEUDOISOTOPY FUNCTORS DISSERTATION ZUR ERLANGUNG DES DOKTORGRADES (DR. RER. NAT.) DER MATHEMATISCH-NATURWISSENSCHAFTLICHEN FAKULTÄT DER RHEINISCHEN FRIEDRICH-WILHELMS-UNIVERSITÄT BONN VORGELEGT VON MALTE MARIO PIEPER AUS BONN BONN 2018 ii Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨atder Rheinischen Friedrich-Wilhelms-Universit¨atBonn 1. Gutachter Prof. Dr. Wolfgang L¨uck 2. Gutacher Prof. Dr. Wolfgang Steimle Tag der Promotion: 18.07.2018 Erscheinungsjahr: 2019 iii Abstract We show the existence of a stable, smooth pseudoisotopyfunctor and construct in the topological, piecewise linear, and smooth category a zig-zag of natural weak equivalences between the stable pseudoiso- topyfunctor and the corresponding functor of Whitehead spectra. iv Contents Introduction vii 1 The Smooth Pseudoisotopy Functor 1 1.1 The Space of Stable Pseudoisotopies . .4 1.1.1 Stereographic projections . .7 1.2 From Continuous Maps to Smooth Embeddings . 17 1.2.1 Strictification of homotopy coherent diagrams . 19 1.2.2 Basic results on manifolds with corners . 22 1.2.3 Smooth approximation . 24 1.2.4 Stable embeddings . 26 1.2.5 Disk bundles and codimension zero embeddings . 27 1.3 Unique Points of Intersection . 29 1.4 The Homotopy Coherent Diagram . 39 1.4.1 Proof of the main theorem . 39 1.4.2 Spectra . 47 1.4.3 All of the δ .......................... 50 2 A natural h{cobordism theorem 51 2.1 Connective Naturality . 53 2.1.1 The stable parametrised h-cobordism theorem . 53 2.1.2 The H{cobordism map . 57 2.1.3 The pseudoisotopy functor . 60 2.1.4 The Whitehead space (1; 1){functor . 70 2.1.5 The natural transformation . 72 2.2 Non-Connective Naturality . 86 2.2.1 Non-connective pseudoisotopies and h{cobordisms . 87 2.2.2 A suspected model for the Whitehead spectrum . 97 2.2.3 The natural transformation . 108 2.2.4 The smooth case . 117 v vi CONTENTS Introduction The automorphism spaces of manifolds have been of major interest to topologists for at least eighty years. The study of automorphism spaces of high-dimensional manifolds as its own field of research came into fruition a little later: Following Stephen Smale's proof of the h{cobordism theorem [42] in 1960, topologists established a flurry of results on the classification of high-dimensional manifolds. In particular, the notion of block bundle automorphisms was intro- duced by Rourke and Sanderson [40] which gave the homotopy fibre sequence of simplicial sets Aut(M) ! Aut(g M) ! Aut(g M)= Aut(M) where Aut(M) and Aut(g M) refers to automorphisms and block bundle auto- morphisms, respectively. The calculation of Aut(g M) via surgery theory proceeded in the following decades. We focus our attention on the quotient Aut(g M)= Aut(M). Let us explain the geometric problem encoded in this quotient: Given some automorphism, we want to decide whether it is isotopic to the Identity. If we follow a strategy similar to the one to figure out whether two given manifolds are isomorphic, then the tools of surgery theory present us with a 1{parameter family of h{cobordisms. The key difference between the two questions is now, that the classification of parametrised families of h{cobordisms is far more subtle than that of h{ cobordisms. Parametrised families of h{cobordisms are just bundles of h{cobordisms with nice clutching maps, hence the problem can be rephrased in understanding the space of possible clutching maps. This is what pseudoisotopies are and how pseudoisotopy spaces enter our story. The study of pseudoisotopies began in earnest in 1970, when Jean Paul Cerf showed his pseudoisotopy theorem [8] which raised the methods of Morse theory, key in the proof of the h{cobordism theorem, to a parametrised setting. In the following years, the study of pseudoisotopy spaces was advanced greatly by, among others, Allen Hatcher, Dan Burghelea, Kyoshi Igusa, Richard Lashof and Friedhelm Waldhausen. In the 1970s, Hatcher [22] described an approach to reduce the geometric questions about pseudoisotopies to a more combinatorial setting. Although his arguments contained several flaws, the general direction of research in the next decades followed along the lines of his ideas. In particular, he claimed that the pseudoisotopy space, intrinsically linked to the automorphism spaces of manifolds, could be connected to the stable vii viii INTRODUCTION pseudoisotopy space which seemed much more accessible via homotopy theory and thus provided a chance for actual computation. Starting in the late 1970s, Waldhausen showed in a series of papers, begin- ning with [49], that this hope was not ill-founded. He introduced A{theory, a functor with good homotopy theoretic properties, and showed that computa- tions about homotopy invariants of stable pseudoisotopy could essentially be reduced to computations on A{theory. In 1987, the same year Waldhausen finished his program (for the most part - we come back to this in a moment), Michael Weiss and Bruce Williams [53] answered precisely how the difference between automorphisms and block bundle automorphisms relates to pseudoisotopies. Let Cat = Top; PL or Diff and let M be a compact Cat{manifold, possibly with boundary. Let Aut@ (M) ⊆ Aut(M) denote the subspace of automorphisms which restrict to the Identity on the boundary, similarly for Autg @ (M). Also, let EZ=2 denote a contractible, free Z=2{CW{complex and P(M) the stable space of Cat{pseudoisotopies relative boundary. Theorem 0.0.0.1 ([53, Theorem A]). There exists a map s 1 2 Φ : Autg @ (M)= Aut@ (M) ! Ω (EZ=2+ ^Z=2 Ω P(M)) which is (k + 1){connected if k is in the Cat{concordance stable range for M. This was only the first article in their ongoing program on combining all of the previous material into a single algebro-topological object to compute the automorphism spaces. Also in 1987, Thomas Farrell and Lowell Jones [15] showed that their meth- ods to simplify h{cobordisms on hyperbolic manifolds via carefully applied geodesic flows and controlled topology [14] could be transferred to pseudoiso- topies. Roughly speaking, their result stated that (for a hyperbolic manifold) each pseudoisotopy was determined by its behaviour on a set of representatives of the isotopy classes of the geodesics. This marked the beginning of what is now known as the Farrell-Jones conjecture for pseudoisotopies. Igusa [30] connected the pseudoisotopy space, intrinsically linked to the au- tomorphism spaces of manifolds, to the stable pseudoisotopy space which by the work of Waldhausen now seemed less mysterious. It also empowered [53, Theorem A]. We denote the unstable pseudoisotopy space by P (M). Theorem 0.0.0.2. The suspension map σ : P Diff (M) ! P Diff (M × [0; 1]) is k{connected, if dim(M) ≥ max(2k + 7; 3k + 4), i.e. the concordance stable range of M is at least k. By now, we also have result for the concordance stable range in the other cases, see [31, Corollary 1.4.2]. As a brief tangent, we note that Goodwillie [18] introduced his calculus of functors in 1990, providing a computation of the differential of pseudoisotopy and thus another tool for actual calculations. After almost two decades, Bjørn Jahren, John Rognes and Friedhelm Wald- hausen [31] gave a proof for the stable parametrised h{cobordism theorem in ix 2005, originally announced by Waldhausen but not published as a part of his original series. In the following theorem, the functors are defined on codimension zero embeddings. Theorem 0.0.0.3 ([31, Theorem 0.1]). There is a natural homotopy equivalence Cat Cat H (M) ' Ω Wh (M) between the stable h{cobordism space and the one fold loops of the Whitehead space for each compact Cat{manifold M. At this point, the results finally provide us with a clear connection between the homotopy groups of the automorphism spaces of high-dimensional manifolds in a certain range and the homotopy groups of A{ and L{theory. Starting with Hatcher, the notion of a pseudoisotopy functor has been part of the discussion and played a rather curious role. Hatcher [23] sketched the definition of a functor on homotopy categories, Burghelea and Lashof [6] pro- vided some necessary details and Waldhausen [50] briefly mentioned it - but did not, in contrast to later references, claim the existence of a strict pseudoisotopy functor on all continuous maps. In this work, we show that a smooth pseudoisotopy functor does indeed exist. This is Theorem 1.4.2.4. Theorem 0.0.0.4. Let Top and Spectra denote the category of compactly gen- erated weak Hausdorff spaces and prespectra, respectively. There is a functor PDiff : Top ! Spectra with the following properties: 1. It descends to a functor of homotopy categories ho PDiff : ho(Top) ! ho(Spectra): 2. There is a natural weak equivalence Ω1PDiff; Spectra !PDiff; Spaces: 3. The subset inclusion PDiff ⊆ PTop extends to a natural transformation of functors of quasicategories. The construction of PTop is given in [12]. As these properties do not uniquely determine the functor, the obvious next step is to compare it with the well-established functor structure on the com- putational side. It turns out that the functors are compatible. This is Corol- lary 2.1.5.18, Theorem 2.2.3.13 and Theorem 2.2.4.2. Theorem 0.0.0.5. Let Cat = Top; PL or Diff. There is a natural weak equiv- alence of (1; 1){functors Ψ: PCat ) Ω2 WhCat;−∞ Cat h:c: h:c: from the (1; 1){functor P : N• Top∆ !N• Spectra∆ of pseudoisotopies to the twofold loops of the functor given by the Whitehead spectrum. In particular, there is a zig-zag of natural weak equivalences between the strict functors PCat : Top ! Spectra and Ω2 WhCat;−∞.