The L2-Cheeger-M¨ullerTheorem for Representations of Hyperbolic Lattices Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN von der KIT-Fakult¨atf¨urMathematik des Karlsruher Instituts f¨urTechnologie (KIT) genehmigte DISSERTATION von Benjamin Waßermann Tag der m¨undlichen Pr¨ufung: 30. Juli 2020 1. Referent: Professor Dr. Roman Sauer 2. Korreferent: Professor Dr. Tobias Lamm 3. Korreferentin: Juniorprofessorin Dr. Beatrice Pozzetti Contents 1 Introduction 4 1.1 Brief history on classical torsion invariants . .4 1.2 L2-torsion on finite CW-complexes and compact manifolds . .7 1.3 L2-torsion on locally symmetric spaces . 10 2 Equivariant Hermitian bundles 16 2.1 The twisted De Rham complex . 16 2.2 Flat bundle isometries . 23 n 2.3 The flat, canonical ρ-bundle over H .............................. 25 3 Analysis on bundles of bounded geometry 33 3.1 Bundles of bounded geometry . 33 3.2 Sobolev spaces . 36 3.3 Uniformly elliptic boundary value problems . 41 3.3.1 The standard boundary value problem on flat bundles . 45 3.4 Applications to the De Rham complex . 47 3.5 Heat kernel estimates . 53 4 Analytic torsion 62 4.1 Hilbert N (Γ)-modules . 62 4.1.1 Hilbert N (Γ)-cochain complexes . 73 4.2 The De Rham and Sobolev complexes . 78 2 4.2.1 Estimates for the spectral density function . 81 4.2.2 Comparison with the combinatorial complex . 84 n 4.2.3 Applications to the representation bundle Eρ # H .................. 85 4.2.4 Proof of large-time convergence . 90 4.3 Small-time convergence . 92 4.3.1 Small-time asymptotics on bundles of bounded geometry . 92 4.3.2 The asymptotic expansion of the hyperbolic heat kernel . 95 5 Combinatorial torsion 101 5.1 Preliminaries . 102 5.2 Cellular torsion of a pair . 103 5.3 Topological torsion . 110 5.3.1 Whitehead torsion of a cochain complex . 111 5.3.2 Whitehead torsion of a homotopy equivalence . 112 5.3.3 L2-Whitehead torsion of a homotopy equivalence . 115 5.4 The Morse-Smale torsion . 122 5.4.1 Unimodular metrics . 133 n 5.5 Applications to the representation bundle Eρ # H ...................... 136 6 The L2-Cheeger-M¨ullertheorem on manifolds with boundary 138 6.1 Preliminaries . 138 6.2 Product formulas, determinant class and subdivisions . 146 6.3 Witten-deformation of the De Rham complex . 151 6.4 Asymptotic expansions . 157 6.5 Proof of Theorem 6.1.5 . 161 6.6 Proof of Corollary C and Theorem E . 165 3 Chapter 1 Introduction The study of torsion invariants, which this thesis is devoted to, has been a driving force in the advance of modern mathematics in the last 100 years. To provide the necessary context, we aim to give a comprehensive, yet by no means exhaustive, summary of important foundational works and key results in that field. Being a prime motivation behind the development and main results of this thesis, the summary will have a strong focus on the close connection between torsion invariants and L2-invariants, as well as the relationship between their respective cellular, topological and analytical versions. At the beginning of the last century, the primary invariants of topological spaces, such as the fundamental group, the (co-)homology groups, as well as derived quantities such as the Betti-numbers and the Euler characteristic, had already been well-known and put to great use. A commonality of all these objects (and simultaneously a reason for their naming) is that they are homotopy invariants, i.e. they agree on homotopy equivalent spaces, and thus they can be used to distinguish between non-homotopy equivalent spaces. As such, they alone are however insufficient in classifying spaces up to more rigid transformations, such as homeomorphisms, diffeomorphisms or isometries. In fact, little was known at that time on elaborate methods that could distinguish between homotopy equivalent, yet non-homeomorphic spaces. 1.1 Brief history on classical torsion invariants The first prominent result in this vein was established by Kurt Reidemeister [80] in 1935. Given a finite simplicial complex K with Ke its universal cover and a unitary representation ρ : π1(K) ! U(n), one can form the twisted cellular cochain complex ∗ ∗ n C (K; ρ) := C (Ke) ⊗ρ C (1.1.1) σ.γ ⊗ v = σ ⊗ ρ(γ)v: Together with a choice of homology bases µ ⊆ H∗(K; ρ), one can define a positive real number T (K; ρ, µ) 2 R>0 nowadays called Reidermeister-Torsion (shortly R-torsion) in honor of its inventor. Provided that ρ is additionally acyclic, i.e. the twisted cohomology H∗(K; ρ) vanishes (so no choice of µ is necessary), it 4 can be computed as follows: Choosing a representative lift for a given oriented basis of cocells in C∗(K), n we obtain linearly independent elements in C∗(Ke). Together with the canonical basis of C , this set of representatives tensors up to become a complex, finite basis for the twisted complex C∗(K; ρ). Choosing the unique inner product on C∗(K; ρ) with respect to which this basis is orthonormal, we can further c p p construct (combinatorial) Laplacians ∆p : C (K; ρ) ! C (K; ρ) for each degree 0 ≤ p ≤ dim(M). Since c ρ was assumed to be acyclic, each ∆p is a positive, invertible endomorphism, and we get an equality dim(M) 1 X log(T (K; ρ)) = (−1)pp log(det(∆c )): (1.1.2) 2 p p=0 Using R-torsion, Reidemeister was able to give a complete combinatorial classification of 3-dimensional lens spaces. While he defined the torsion only for complexes on lens spaces, Franz [37] generalized the definition of R-torsion to arbitrary finite simplicial complexes. In the same process, he generalized Reidemeister's classification result onto higher-dimensional lens spaces. Both authors implicitly used the important fact that the R-torsion associated to acyclic representations is a combinatorial invariant: Two simplicial structures of the same space admitting isomorphic subdivisions have the same acyclic R-torsion, a fact that was later proven rigorously in a more general fashion by Whitehead in a series of papers [100], [99], [98], [97]. Namely, he proved that R-torsion is invariant under a special class of homotopy equivalences, so-called simple homotopy equivalences. In the same body of work, he also gave the homotopy classification of lens spaces and provided examples of homotopy equivalent lens spaces with different R-torsions, thus showing that not every homotopy equivalence is simple. All of this he achieved by introducing a new torsion invariant, called the Whitehead torsion, which provided an effective new tool in analyzing two homotopy equivalent spaces. Lastly, he also raised the question whether all homeomorphims are simple homotopy equivalences. Combined with his previous result, this would imply that R-torsion for acyclic representations is a homeomorphism invariant. Arguably the decade in which Whitehead torsion was used most beneficially, the 1960's began with a satisfying completion of the earlier mentioned classification results by Reidemeister, Franz and Whitehead, obtained by Brody [17] in 1960. Namely, he proved that the simple homotopy class, the combinatorial class and the homeomorphism class of lens spaces all agree. Shortly afterwards in 1961, Milnor [67] prominently applied a relative version of R-torsion to disprove the "Hauptvermutung", one of the famous topological conjectures of its time which stated that two homeomorphic finite-dimensional simplical complexes always would always have isomorphic subdivisions. He was also the one to define another torsion invariant, called the Milnor torsion, which would generalize Whitehead torsion even further. Thirdly (and perhaps most famously), Whitehead torsion played an important part in the proof of the s-cobordism theorem, shown independently by Mazur [65], Stallings [93], and Barden [5] for the category of piecewise-linear and smooth manifolds, later extended to the category of topological manifolds by Kirby and Siebenmann [49, Essay II]. It says that for n ≥ 5, given a topological/PL/smooth inclusion M n ,! N n+1 of a closed n-manifold into a compact (n+1)-manifold that is a homotopy equivalence, one has N =∼ M ×[0; 1] (here, =∼ stands for the isomorphism in the respective category) if and only if the inclusion is a simple homotopy equivalence. Most notably, the s-cobordism theorem has as an (almost) immediate consequence the generalized Poincar´e conjecture for dimensions n ≥ 5: Any n-manifold homotopy equivalent to the n-sphere Sn must in fact be homeomorphic to Sn. To close off the decade, it was shown by Kirby and Siebenmann [50] in 1969 for simplicial structures on manifolds, later by Chapman [24] in 1975 for arbitrary simplicial complexes, that all homeomorphisms are simple homotopy equivalences, thereby confirming Whitehead's old conjecture that R-torsion for acyclic representations indeed is a homeomorphism invariant. 5 Among the most influential mathematical achievements that were brought forth in the 60's also ranks the Atiyah-Singer index theorem. Roughly stated, this deep result relates the index of certain elliptic differential operators defined over a manifold M to various of its topological properties. In the aftermath of the proof, more and more people became interested in discovering new sophisticated analytic quantities on manifolds that could possibly shed light on equally sophisticated topological properties. One of the quantities discovered this way was the analytic torsion, first appearing in a paper by Ray and Singer [78] in 1971, where they rigorously defined it and proved some of its basic properties (therefore, it is sometimes called Ray-Singer torsion): Given a closed Riemannian manifold (M; g), and a unitary representation ρ : π1(M) ! U(n), one obtains the flat, complex vector bundle Eρ # M over M associated to ρ, with Eρ the total space of the bundle and M the base space (this notation will be used throughout the thesis).