Algebraic K-theory of the group ring of the fundamental group

Question

I know of two places where $K_{*}(\mathbb{Z}\pi_{1}(X))$ (the algebraic $K$-theory of the group ring of the fundamental group) makes an appearance in algebraic topology.

The first is the Wall finiteness obstruction. We say that a space $X$ is finitely dominated if $id_{X}$ is homotopic to a map $X \rightarrow X$ which factors through a finite CW complex $K$. The Wall finiteness obstruction of a finitely dominated space $X$ is an element of $\tilde{K_{0}}(\mathbb{Z}\pi_{1}(X)) $ which vanishes iff $X$ is actually homotopy equivalent to a finite CW complex.

The second is the Whitehead torsion $\tau(W,M)$, which lives in a quotient of $K_{1}(\mathbb{Z}\pi_{1}(W))$. According to the s-cobordism theorem, if $(W; M, M')$ is a cobordism with $H_{*}(W, M) = 0$, then $W$ is diffeomorphic to $M \times [0, 1]$ if and only if the Whitehead torsion $\tau(W, M)$ vanishes.

For more details, see the following:

http://arxiv.org/abs/math/0008070 (A survey of Wall's finiteness obstruction)

http://www.maths.ed.ac.uk/~aar/books/surgery.pdf (Algebraic and Geometric Surgery. See Ch. 8 on Whitehead Torsion)

My question is twofold.

First, is there a high-concept defense of $K_{*}(\mathbb{Z}\pi_{1}(X))$ as a reasonable place for obstructions to topological problems to appear? I realize that $\mathbb{Z}\pi_{1}(X)$ appears because the (cellular, if $X$ is a cell complex) chain groups of the universal cover $\tilde{X}$ are modules over $\mathbb{Z}\pi_{1}(X)$. Is it the case that when working with chain complexes of $R$-modules, we expect obstructions to appear in $K_{*}(R)$?

Second, is there an enlightening explanation of the formal similarity between these two obstructions? (Both appear from considering the cellular chain complex of a universal cover and taking an alternating sum.)

Answer 1

To add to Tim Porter's excellent answer:

The story of what we now call $K_1$ of rings begins with Whitehead's work on simple homotopy equivalence, which uses what we now call the Whitehead group, a quotient of $K_1$ of the group ring of the fundamental group of a space.

On the other hand, the story of $K_0$ of rings probably begins with Grothendieck's work on generalized Riemann-Roch. What he did with algebraic vector bundles proved to be a very useful to do with other kinds of vector bundles, and with finitely generated projective modules over a ring, and with some other kinds of modules.

I don't know who it was who recognized that these two constructions deserved to be named $K_0$ and $K_1$, and viewed as two parts of something larger to be called algebraic $K$-theory. But Milnor gave the right definition of $K_2$, and Quillen and others gave various equivalent right definitions of $K_n$.

Let me try to lay out the parallels between the topological significances of Whitehead's quotient of $K_1(\mathbb ZG)$ and Wall's quotient of $K_0(\mathbb ZG)$. My main point is that both of them have their uses in both the theory of cell complexes and the theory of manifolds.

The Whitehead group of $G$ is a quotient of $K_1(\mathbb ZG)$. Its significance for cell complexes is that it detects what you might call non-obvious homotopy equivalences between finite cell complexes. An obvious way to exhibit a homotopy equivalence between finite complexes $K$ and $L$ is to by attaching a disk $D^n$ to $K$ along one half of its boundary sphere and obtain $L$. Roughly, a homotopy equivalence between finite complexes is called simple if it is homotopic to one that can be created by a finite sequence of such operations. The big theorem is that a homotopy equivalence $h:K\to L$ between finite complexes determines an element (the torsion) of the Whitehead group of $\pi_1(K)$, which is $0$ if and only if $h$ is simple, and that for any $K$ and any element of its Whitehead group there is an $(L,h:K\to L)$, unique up to simple homotopy equivalence, leading to this element in this way, and that this invariant of $h$ has various formal properties that make it convenient to compute.

One reason why you might care about the notion of simple homotopy equivalence is that for simplicial complexes it is invariant under subdivision, so that one can in fact ask whether $h$ is simple even if $K$ and $L$ are merely piecewise linear spaces, with no preferred triangulations. This means that, for example, a homotopy equivalence between compact PL manifolds (or smooth manifolds) cannot be homotopic to a PL (or smooth) homeomorphism if its torsion is nontrivial. (Later, the topological invariance of Whitehead torsion allowed one to eliminate the "PL" and "smooth" in all of that, extending these tools to, for example, topological manifolds without using triangulations.)

But the $h$-cobordism theorem says more: it applies Whitehead's invariant to manifolds in a different and deeper way.

Meanwhile on the $K_0$ side Wall introduced his invariant to detect whether there could be a finite complex in a given homotopy type. Note that where $K_0$ is concerned with existence of a finite representative for a homotopy type, $K_1$ is concerned with the (non-)uniqueness of the same.

Siebenmann in his thesis applied Wall's invariant to a manifold question in a way that corresponds very closely to the $h$-cobordism story: The question was, basically, when can a given noncompact manifold be the interior of a compact manifold-with-boundary? Note that there is a uniqueness question to go with this existence question: If two compact manifolds $M$ and $M'$ have isomorphic interiors then this leads to an $h$-cobordism between their boundaries, which will be a product cobordism if $M$ and $M'$ are really the same.

One can go on: The question of whether a given $h$-cobordism admits a product structure raises the related question of uniqueness of such a structure, which is really the question of whether a diffeomorphism from $M\times I$ to itself is isotopic to one of the form $f\times 1_I$. This is the beginning of pseodoisotopy theory, and yes $K_2$ comes into it.

But from here on, the higher Quillen $K$-theory of the group ring $\mathbb Z\pi_1(M)$ is not the best tool. Instead you need the Waldhausen $K$-theory of the space $M$, in which basically $\mathbb Z$ gets replaced by the sphere spectrum and $\pi_1(M)$ gets replaced by the loopspace $\Omega M$. It's a long story!

Answer 2

First a slight quibble: the Whitehead group is the origin of algebraic K-theory, and predates the general stuff by quite a time, so your wording is not quite fair to Whitehead!

On your first question, one direction to look is at Waldhausen's K-theory. (The original paper is worth studying, but you should also look at the more recent stuff relating to the connections between that and model categories.) Waldhausen has a section on the links between his groups and the Whitehead simple homotopy theory. (I can provide some references if you need them. There are some useful comments in the nLab if you search on 'Waldhausen'.)

For the second question, it is probably a good idea to look at some books on Simple Homotopy Theory and to take a slightly historical perspective, (also to glance at the original articles not just surveys). Whitehead's and then Milnor's papers released a set of tools for studying finite CW-complexes. The role of the chains on the universal cover can be viewed in various ways, but both constructions are part of the general idea at the time of making homotopy theory more 'constructive' and the taming of the fundamental group action in non-Abelian cases was a first step.

The origins of simple homotopy theory are pre-world war II with Riedemeister and note that Whitehead's two part paper in 1949 was called 'Combinatorial Homotopy Theory' and was intended to mirror 'Combinatorial Group Theory'.

This does not give the enlightenment on the 'reasons' for the similarity but may help you to gain some knowledge of the origins of that stuff and sometimes that helps to see what side alleys have been left unexplored and to provide an overview of the area. I have a sneaky idea that there is a K-theory for homotopy 2-types (and beyond) and that the Waldhausen construction is a way into that, but I may be wrong on that.

On a completely different tack, have a look at Kapranov and Saito's article:

Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions , in Higher homotopy structure in topology and mathematical physics (Poughkeepsie, N.Y. 1996) , volume 227 of Contemporary Mathematics , 191–225

This links up the Steinberg group (which has a neat motivation from linear algebra) with a lot of homotopical and homological algebra. I will not say more on this as this is already getting a bit long, but do ask if you find these references useful but need further pointers.