A-Theory of Spaces

Waldhausen's A-theory $A(X)$ is the algebraic K-theory of the category of spaces "parametrized" over $X$ . It provides a homotopy-theoretic approach to studying the space of h-cobordisms and diffeomorphisms of manifolds, connecting algebraic K-theory to geometric topology.

Definition

Definition7.3A-theory

For a topological space $X$ (or simplicial set), the A-theory space $A(X)$ is defined as the K-theory of the Waldhausen category $\mathcal{R}(X)$ of retractive spaces over $X$ :

Objects: spaces $Y$ with maps $X \xrightarrow{s} Y \xrightarrow{r} X$ satisfying $r \circ s = \operatorname{id}_X$ , where $Y$ is homotopy equivalent to a finite CW complex relative to $X$ .
Cofibrations: maps $Y_1 \to Y_2$ over $X$ that are cofibrations of spaces.
Weak equivalences: maps $Y_1 \to Y_2$ over $X$ that are homotopy equivalences.

Then $A(X) = K(\mathcal{R}(X)) = \Omega |wS_\bullet \mathcal{R}(X)|$ .

The A-theory groups are:

$A_n(X) = \pi_n(A(X)) \quad \text{for } n \geq 0.$

ExampleA-theory of a point

For $X = *$ (a point), $\mathcal{R}(*)$ is the category of finite pointed CW complexes (retractive over $*$ = basepoint). The K-theory is:

$A(*) \simeq \mathbb{Z} \times B\widetilde{GL}(\mathbb{S})^+$

where $\mathbb{S}$ is the sphere spectrum and $\widetilde{GL}(\mathbb{S})$ is the "general linear group" of the sphere spectrum (automorphisms of wedges of spheres, up to stable equivalence).

There is a natural linearization map $A(*) \to K(\mathbb{Z})$ induced by taking the integral homology of a space:

$\mathcal{R}(*) \to \operatorname{Perf}(\mathbb{Z}), \quad Y \mapsto C_*(Y; \mathbb{Z}).$

This map is not an equivalence: the fiber $\operatorname{Wh}^{\text{Diff}}(*)$ contains information about exotic smooth structures.

Connection to h-cobordisms

Definition7.4The Whitehead space

The Whitehead space $\operatorname{Wh}(X)$ fits into a fibration sequence:

$X_+ \wedge A(*) \to A(X) \to \operatorname{Wh}(X)$

where $X_+$ is $X$ with a disjoint basepoint and the first map is the "assembly." Alternatively:

$\operatorname{Wh}(X) = \Omega^{\infty}(\mathbf{A}(X) / (X_+ \wedge \mathbf{A}(*)))$

where $\mathbf{A}(X)$ is the A-theory spectrum.

The fundamental connection to topology is: for a compact smooth manifold $M$ of dimension $\geq 5$ :

$\operatorname{Wh}(M) \simeq \Omega^2 \mathcal{H}(M)$

where $\mathcal{H}(M)$ is the space of h-cobordisms on $M$ (Waldhausen's theorem). Thus $\pi_0(\mathcal{H}(M)) \cong \pi_2(\operatorname{Wh}(M))$ , and the h-cobordism theorem says this is trivial when $\pi_1(M) = 0$ (since $\operatorname{Wh}(\pi_1) = 0$ ).

ExampleA-theory of the circle

For $X = S^1$ with $\pi_1 = \mathbb{Z}$ :

$A(S^1) \simeq A(*) \times \operatorname{hofib}(1 - \sigma: A(*) \to A(*))$

where $\sigma$ is the "shift" automorphism corresponding to the deck transformation. This is related to the Nil-groups of Bass: the fiber of the assembly map contains the Nil-K-theory.

The Whitehead space $\operatorname{Wh}(S^1)$ is related to the pseudoisotopy space of the circle and computes the stable space of diffeomorphisms:

$\pi_0(\operatorname{Diff}(M)) \text{ (stable)} \sim \pi_*(\operatorname{Wh}(M))$

for high-dimensional manifolds $M$ .

The linearization map

RemarkA-theory vs K-theory of group rings

The linearization map $L: A(X) \to K(\mathbb{Z}[\pi_1(X)])$ (for connected $X$ ) is induced by:

$\mathcal{R}(X) \to \operatorname{Perf}(\mathbb{Z}[\pi_1(X)]), \quad Y \mapsto C_*(\tilde{Y}; \mathbb{Z})$

where $\tilde{Y}$ is the universal cover and $C_*$ denotes cellular chains, viewed as a complex of $\mathbb{Z}[\pi_1]$ -modules.

At the level of $\pi_1$ : the map $\pi_1(A(X)) \to K_1(\mathbb{Z}[\pi_1])$ sends the "Whitehead torsion" of an h-cobordism to its algebraic Whitehead torsion in $\operatorname{Wh}(\pi_1)$ .

The relative term $\operatorname{fib}(L)$ carries the "smooth" information beyond the algebraic K-theory:

$\pi_0(\operatorname{fib}(L)) = 0$ (the Whitehead torsion is a complete obstruction in the s-cobordism theorem).
$\pi_k(\operatorname{fib}(L))$ for $k > 0$ detects exotic smooth structures and "higher Reidemeister torsion."

Stable parametrized h-cobordism theorem

Theorem7.2Waldhausen's Stable Parametrized h-Cobordism Theorem

For a compact smooth manifold $M$ of dimension $d$ , let $\mathcal{H}(M)$ denote the space of smooth h-cobordisms on $M$ and $\mathcal{H}^s(M) = \operatorname{colim}_k \mathcal{H}(M \times D^k)$ the stable h-cobordism space. Then:

$\mathcal{H}^s(M) \simeq \Omega \operatorname{Wh}(M).$

This identifies:

$\pi_0(\mathcal{H}^s(M)) \cong \pi_1(\operatorname{Wh}(M)) = \operatorname{Wh}_1(\pi_1(M))$ (the classical Whitehead group).
$\pi_k(\mathcal{H}^s(M)) \cong \pi_{k+1}(\operatorname{Wh}(M))$ for all $k \geq 0$ .

Combined with the assembly map: the stable h-cobordism space is controlled by the algebraic K-theory of $\mathbb{Z}[\pi_1(M)]$ and the difference between A-theory and K-theory.

Exampleh-cobordisms on spheres

For $M = S^n$ ( $n \geq 5$ ), $\pi_1 = 0$ , so $\operatorname{Wh}(\pi_1) = 0$ and $\pi_0(\mathcal{H}(S^n)) = 0$ (the h-cobordism theorem). But $\pi_k(\mathcal{H}^s(S^n))$ for $k > 0$ is generally nontrivial and is computed by:

$\pi_k(\mathcal{H}^s(S^n)) \cong \pi_{k+1}(\operatorname{Wh}(*)) \cong \pi_{k+1}(\operatorname{fib}(A(*) \to K(\mathbb{Z})))$

For example, $\pi_1(\mathcal{H}^s(S^n)) \cong \pi_2(\operatorname{Wh}(*))$ detects exotic smooth structures on $h$ -cobordisms -- not on $S^n$ itself, but on families of h-cobordisms parametrized by $S^1$ . The groups $\pi_*(\operatorname{Wh}(*))$ are computed in terms of stable homotopy theory and are related to the image of the J-homomorphism.