The aim of the talk is to define an invariant (« universal \( L^2 \)-torsion ») from which many others (usual \( L^2 \)-torsion, \( L^2 \)-Alexander invariant and Euler characteristic,…) can be derived, as well as the relations between them.

Notation

• The group \( K_1({\mathbb Z} G) \) is the abelian group generated by all invertible matrices \( A \in M_n({\mathbb Z} G), n \ge 0 \) with relations \( [AB] = [A] + [B] \) and \( \left[\begin{pmatrix} A & C \\ 0 & B \end{pmatrix} \right] = [A] + [B] \).
• The Whitehead group \( \mathrm{Wh}(G) \) is the quotient of \( K_1({\mathbb Z} G) \) by the elements \( [(\pm g)], g \in G \).
• The weak \( K_1 \)-group \( K_1^w(G) \) is defined as \( K_1(G) \) but by taking the larger set of matrices which define « weak isomorphisms » (have no kernel in \( L^2(G) \)).

Universal \( L^2 \)-torsion

Let \( C_* \) be a finite complex of free, finitely generated \( {\mathbb Z} G \)-complexes, each \( C_i \) having a preffered basis. If \( C_* \) is acyclic, that is \( H_*(C_*) = 0 \), then the Reidemeister torsion \( \rho(C_*) \in K_1({\mathbb Z} G) \) is well-defined (and independent of the choice of basis). If it is only \( L^2 \)-acyclic then there is a well-defined invariant \( \rho_u^{(2)}(C_*) \in K_1^w({\mathbb Z} G) \) which has a universal property.

For the rest of the talk use the following convention: \( X, Y, \ldots \) will be finite CW-complexes, and \( \overline X, \overline Y, \ldots \) will be \( G \)-covers of each of them. If \( H_*^{(2)}(\overline X; G ) = 0 \) then denote by \( \rho_u^{(2)}(\overline X) = \rho_u^{(2)}(C_*(\overline X)) \).

Properties of the universal \( L^2 \)-torsion

• Homotopy invariance: Let \( f: \overline X \to \overline Y \) be a \( G \)-equivariant homotopy equivalence, and let \( \rho(f) \in K_1({\mathbb Z} G) \) be its Whitehead torsion. Then \( \rho(f) \) maps to \( \rho_u^{(2)}(\overline X) – \rho_u^{(2)}(\overline X) \) under the natural map \( K_1({\mathbb Z} G) \to K_1^w({\mathbb Z} G) \).
• Additivity: The universal torsion is additive in exact sequences; if the inclusion maps between CW-complexes are \( \pi_1 \)-injective then everything is functorial with respect to maps from the classical \( K_1 \).
• Fibrations: If \( F \overset{i_*}{\to} E \to B \) is a fibration such that \( i_* \) is \( pi_1 \)-injective then \( \rho_u^{(2)}(E) = \chi(B)i_*\rho_u^{(2)}(\overline F) \).
• Poincaré duality: If \( X \) is a \( n \)-dimensional manifold then \( \rho_u^{(2)}(X) = (-1)^{n+1}\rho^{(2)}(X)^* \).
• Realisability: If there exists an \( L^2 \)-acyclic \( G \)-universal cover then for all \( \eta \in K_1^w({\mathbb Z} G) \) there exists an \( L^2 \)-acyclic covering space \( \overline X \) with \( \rho_u^{(2)}(\overline X) = \eta \).
• 3–manifolds: The universal \( L^2 \)-torsion can be computed from the geometric pieces, and for manifolds with toric boundary it is equal to \( [A_i] – [s_i-1] \) where \( A \) is the Fox matrix minus its i-th column (and \( s_i \) the corresponding generator).

An example

Suppose \( G = {\mathbb Z} \); then \( \mathrm{Wh}(G) = \{ 0 \} \) and \( \mathrm{Wh}^w(G) = {\mathbb Q}(x) \). The universal \( L^2 \)-torsion is the Reidemeister torsion.

\( L^2 \)-torsion and the \( L^2 \)-Alexander invariant

The Fuglede–Kadison determinant gives a morphism \( \mathrm{Wh}^w(G) \to {\mathbb R}_{\ge 0} \). The image of \( \rho_u^{(2)}(\overline M) \) under this map is the \( L^2 \)-torsion \( \rho^{(2)}(\overline M) \).

Note that if \( M \) is a compact, even-dimensional manifold the \( L^2 \)-torsion is always trivial by Poincaré duality, while the universal \( L^2 \)-torsion might not be.

If \( G \) is resudually finite, then for every cohomology class \( \phi \in H^1(M) \) there is associated a map \( {\mathbb R}_{>0} \to {\mathbb R}_{\ge 0} \), the \( L^2 \)-Alexander torsion \( \rho^{(2)}(\overline M, \phi) \). It can be constructed through the universal _( L^2 \)-torsion; a more general hope is that there exists a pairing
\[
\mathrm{Rep}(G) \times \mathrm{Wh}^w(G) \to {\mathbb R}.
\]

There is also a map from \( \mathrm{Wh}^w(G) \to \mathcal P(H_1(G)_{\mathrm{free}}) \), the space of polytopes in the first homology space with integral vertices. This associates to a \( G \)-covering space a polytope \( \mathrm P(\overline M) \).

Theorem: For a closed 3–manifold \( M \), \( \mathrm P(\widetilde M) \) is roughly the unit polytope fr the Thurston norm of \( M \).

\( L^2 \)-Euler characteristic and twisted version

If \( C_* \) is any \( {\mathbb Z} G \) chain complex, consider the \( \mathcal NG \)-cchain complex \( C_* \otimes_{{\mathbb Z} G} \mathcal NG \). It has well-defined (possibly infinite) \( L^2 \)-Betti numbers defined using the following result.

Theorem (Lück, XX^th century): There exists a dimension function for arbitrary \( \mathcal NG \)-modules, which extends the von Neumann dimension.

Suppose that the series \( \sum_{n \in {\mathbb Z}} b_n^{(2)}(C_*) \) converges. Let \( h^{(2)}(C_*) \) be its sum, and define the \( L^2 \)-Euler characteristic by \( \chi^{(2)}(C_*) = \sum_{n\in{\mathbb Z}} (-1)^n b_n^{(2)}(C_*) \).

Let \( \Phi: G \to {\mathbb Z} \). Define
\[
C_*^\Phi(\overline X) = C_*(\overline X) \otimes_{\mathbb Z} \phi^*{\mathbb Z}[{\mathbb Z}]
\]
and let
\[
h^{(2)}(\overline X ; \Phi) = h^{(2)}(C_*^\Phi(\overline X)), \: \chi^{(2)}(\overline X; \Phi) = \chi^{(2)}(C_*^\Phi(\overline X)).
\]
Note that if \( K = \ker(\Phi) \) then \( b_n^{(2)}(C_*^\Phi(\overline X)) = b_n^{(2)}(\mathcal NK \otimes_{{\mathbb Z} K} C_*|_{{\mathbb Z} K}) \).

Theorem: Suppose that \( G \) is torsion-free and satisfies the Atiyah conjecture. Then \( h^{(2)}(\overline X; \Phi) \) is finite and \( \chi^{(2)}(\overline X; \Phi) \) is an integer.

Example

Let \( E \) be the mapping torus of an homeomorphism \( f : F \to F \) and \( p: E \to \mathbb S^1 \) the natural projection. Let \( \Phi = p_* \) and \( \overline E \) the associated infinite cyclic covering space, which is homotopy equivalent to \( F \). Then:
\[
\chi^{(2)}(\widetilde E; \Phi) = \chi^{(2)}(\mathcal NK) \otimes_{{\mathbb Z} K} C_*(\widetilde E)|_K) = \chi^{(2)}(\mathcal NK \otimes C_*(F))
\]
so in the end \( \chi^{(2)}(\widetilde E; \Phi) = \chi(F) \).

In general, the following result is valid.

Theorem (Friedl–Lück): If \( M \) is any closed 3–manifold then \( \chi^{(2)}(\widetilde M; \Phi) = \|\Phi\|_T \).

An application to knot theory

Question (Simon): Let \( \mathbb S^3 \setminus K, \mathbb S^3 \setminus K’ \) be two knots in teh 3–sphere and \( M = \mathbb S^3 \setminus K, M’ = Y’ \). Suppose that there exists a continuous map \( f: M \to M’ \) such that \( \pi_1(f) \) is surjective and \( f_* \) is an isomorphism \( H_1(M; {\mathbb Q}) \to H_1(M'; {\mathbb Q}) \). Does it follow that \( g(K) \ge g(K’) \)?

It can be shown that if \( G \) is residually elementary amenable locally indicable then \( \chi^{(2)}(\widetilde M; f^*\phi) \ge \chi^{(2)}(\widetilde{M’}; \phi) \), which implies a positive answer by the theorem above.