# Can anyone explain to me what is an assembly map?

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.

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.

• A miniqueston: what is the motivation for the term assembly? – Mariano Suárez-Álvarez Mar 2 '12 at 20:11
• I am not sure, but I believe it has to do with the fact that the domain is local and the codomain is global. Thus the assembly map is the passage from local data to global data and this passage is given by "assembling" the local information. More precisely, The domain of the assembly map is given by "assembling" the value of the functor at simplices over $X$ = "thickened up points" in $X$ in a compatible way (=hocolim), and mapping the resulting homology theory into the global object. – John Klein Mar 2 '12 at 22:19

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.