VyperTD » The Plus construction

The Plus construction

This is going to be slightly uncomfortable for people who just want to know what I’ve been doing this summer, but this post is going to be about Quillen’s Plus construction. Why? Because I was trying to remember the details when I was having dinner in Bar Uno the other night, and couldn’t quite get it all out. Also I want to test out my new $\small \color{white}LaTeX$ renderer, which incidentally looks a bit bad as it is giving images to put on a white background, and has to be recompiled to change that, and I don’t have shell access yet. Incidentally, after looking up the bible, as mine was too heavy to bring over from Oxford, I notice the Allen Hatcher/Terri Hatcher Google war is being dominated by the topologist. Good stuff.

The Why - Quite often in algebraic topology, having a nontrivial action of the fundamental group on the higher homotopy groups make syour life strictly more difficult than it needs to be, so wouldn’t it be nice to have a way to trivialise the action? In general this would be complicated, but if we can find a related space that is simply-connected then the action is trivial by default. So let $\small \color{white}X$ be a CW-complex.
The How - By Hurewicz’s theorem, we have homomorpisms $\small \color{white}h:\pi_i \to H_i$ . In particular we have $\small \color{white}h:\pi_1 \to H_1$ which can be shown to have kernel $\small \color{white}\pi_1^\prime$ (notice then that if $\small \color{white}H_1$ is trivial then $\small \color{white}\pi_1$ must be perfect). Suppose then that we are lucky enough to have trivial $\small \color{white}H_1$ . Choose loops $\small \color{white}\varphi_\alpha$ generating $\small \color{white}\pi_1$ , and use them to attach 2-cells $\small \color{white}e_\alpha^2$ to $\small \color{white}X$ to get a simply-connected CW-complex $\small \color{white}X^\prime$ . So far so good, but this will have ruined the homology. The long exact sequence for homology of the pair $\small \color{white}(X^\prime, X)$ now gives

$\small \color{white}0 \to H_2(X) \to H_2(X^\prime) \to H_2(X^\prime, X) \to 0$

and the last term is free-abelian on the 2-cells $\small \color{white}e_\alpha^2$ so this splits to give $\small \color{white}H_2(X^\prime) \simeq H_2(X) \oplus H_2(X^\prime, X)$ . Since $\small \color{white}X^\prime$ is simply-connected, Hurewicz theorem gives $\small \color{white}\pi_2(X^\prime) \simeq H_2(X^\prime)$ so we can represent a basis of the free direct summand $\small \color{white}H_2(X^\prime, X)$ with maps $\small \color{white}\psi_\alpha : S^2 \to X^\prime$ , which we can also assume to be cellular, and so attach 3-cells $\small \color{white}e_\alpha^3$ via them to $\small \color{white}X^\prime$ , to get a simply-connected CW complex $\small \color{white}X^+$ .

Now $\small \color{white}X^+ / X$ is a wedge of $\small \color{white}S^2$ ’s with 3-cells attached inside them, and so is contractible. Thus $\small \color{white}H_i(X^+, X) = 0$ for $\small \color{white} i > 0$ and both are connected, so the inclusion $\small \color{white}X \to X^+$ induces an isomorphism on homology, and $\small \color{white}X^+$ is a simply-connected CW complex, as required.