Can anyone explain to me what is an assembly map?

Question

Or can you give me a good place to read about things related to assembly map, besides wikipedia? I am specially interested in the case of algebraic K-theory. Would appreciated if you could provide examples.

Answer 1

If you are given a homotopy functor $L$ from spaces to spectra, the assembly map is a natural map of spectra $$ H_\bullet(X;L) \to L(X) , $$ where the domain is a homology theory. This homology theory is the "homology of $X$ with coefficients in $L$," that is, $X_+ \wedge L(\text{pt})$.

This map is a universal approximation to $L$ on the left by a homology theory in the homotopy category of functors.

The map can most easily be described as follows: let $\Delta_X$ be the category whose objects are maps $\sigma: \Delta^n \to X$ (where $n$ can vary) and whose morphisms are given by restricting to a face of $\Delta^n$. Then there is an evident map $$ \underset{\sigma\in \Delta_X}{\text{hocolim}} L(\Delta^n) \to L(X) $$ where the map $L(\Delta^n) \to L(X)$ is given by applying $L$ to $\sigma$. The domain of this map is a homology theory, and in fact it's the homology of $X$ with coefficients in $L$.

Notes

1. Algebraic K-theory is the case of the functor $K: X \mapsto K(\Bbb Z[\pi(X)])$ where $\pi(X)$ denotes the fundamental groupoid. If we restrict to $X = B\pi$ for $\pi$ a discrete group, we get $H_\bullet(B\pi;K) \to K(\Bbb Z[\pi])$.

2. An important case is that of surgery theory. In that case, the construction needs to be modified slightly: the category of spaces needs to be replaced with the category of spaces equipped with stable spherical fibration (this is the over-category $TOP_{/BG}$, where $BG$ classifies stable spherical fibrations).

3. In the surgery theory case, if $X = B\pi$, the Novikov conjecture can be stated: it says that the assembly map is a split surjection on rational homotopy. The Borel conjecture states that the assembly map is a weak equivalence. Of course, these statements are only known to be true under certain assumptions.

4. When $L = A$ is Waldhausen's algebraic $K$-theory of spaces functor, the homotopy fiber of the assembly map at a manifold $X$ is the moduli space for the $h$-cobordisms relative to $X$ after a suitable stabilization with respect to dimension. This is a very deep result, the details of which have only just recently been written down by Jahren, Rognes and Waldhausen.

5. The assembly map was first due to Quinn. The formulation I gave appears in a paper by Weiss and Williams: Assembly, in Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), 332--352, London Math. Soc. Lecture Notes 227, Cambridge Univ. Press, Cambridge, 1995.

6. Here's the reason that the domain of the map constructed above is a homology theory: there is a natural transformation $$ \underset{\sigma\in \Delta_X}{\text{hocolim}} L(\Delta^n) \to \underset{\sigma\in \Delta_X}{\text{hocolim}} L(\text{pt})\, . $$ As $L(\Delta^n) \to L(\text{pt})$ is a weak equivalence (since $L$ is a homotopy functor), the same is true after taking hocolim. Since the second of these hocolims is that of a constant functor, it's given by $|\Delta_X|_+ \wedge L(\text{pt})$ (this is a standard exercise). Finally, the realization of the nerve, $|\Delta_X|$ is weak equivalent to $X$, also by a standard argument.

Answer 2

You can find lots of definitions in Hambleton and Pedersen's paper "Identifying assembly maps in K- and L-theory " at http://www.maths.ed.ac.uk/~aar/papers/idensurg.pdf

For a torsion-free group $G$, the assembly map $H_n(BG;K\mathbb{Z})\rightarrow K_n(\mathbb{Z}G))$ is induced by the contraction $EG\rightarrow pt$. In General, the algebraic $K$-theory is difficult to compute. The Farrell-Jones conjecture (claiming the assembly map is an isomorphism for each $n$) says that the algebraic $K$-theory of group rings can be computed by a homology theory, which is easier to deal with.