# how to prove the $n$-times self-product of a map is null-homotopic

Let $k$ be a fixed positive integer and $\Sigma_k$ the $k$-th symmetric group. By letting $\Sigma_k$ permuting an orthonormal basis of a $k$-dimensional Euclidean space, there is a "regular representation" $$r_k: \Sigma_k\longrightarrow O(k)$$ which induces a map between classifying spaces $$\rho_k: B\Sigma_k\longrightarrow BO(k).$$ Let $X$ be a finite CW-complex and a map $$f: X\longrightarrow B\Sigma_k.$$ For any positive integer $n$, we produce a map by taking self-product of $f$ for $n$-times $$\prod_n f: X\longrightarrow \prod_n B\Sigma_k$$ and have a composition $$g_n: X\overset{\prod_n f}{\longrightarrow}\prod_n B\Sigma_k\overset{\prod_n \rho_k}{\longrightarrow}\prod_n BO(k)\longrightarrow BO(nk)=G_{nk}(\mathbb{R}^\infty).$$ where the last map is induced by the inclusion $$\prod_n O(k)\longrightarrow O(nk).$$ Question: I want to prove that for any $k$, any finite CW-complex $X$ and any map $f$, there exists a positive integer $n$ such that $g_n$ is null-homotopic. Is it true? How to prove?

Yes, this is true. Your map $B\Sigma_k\rightarrow BO(k)$ gives rise to a map $B\Sigma_k\rightarrow BO$, i.e. an element in the reduced K-theory group $\tilde{ko}^0(B\Sigma_k)$.

Now that whole group is probably not torsion, but the Atiyah-Hirzebruch spectral sequence tells you that, if $K$ is a finite-dimensional $CW$ complex with torsion homology groups, $\tilde{ko}^0(K)$ is torsion.

So if you take $K$ to be a suitably modified skeleton of $B\Sigma_k$ (see below), it follows that there is $n$ such that the map $$K\rightarrow B\Sigma_k\rightarrow BO(k)\rightarrow BO(nk)$$ is null homotopic, where the last map is the times $n$ map.

But if $X$ is a finite-dimensional $CW$-complex, the map $X\rightarrow B\Sigma_k$ will factor over $K$ if we choose $K$ to be a sufficiently high-dimensional "skeleton".

I was imprecise above about $K$ being a "suitably modified skeleton", let me clarify this:

If $Y$ is a CW-complex with torsion homology groups, then, for each $n$, there is an $n+1$-dimensional CW complex $K$ with torsion homology groups and a map $K\rightarrow Y$ that induces isomorphisms on homology up to degree $n-1$.

To build that $K$, simply take $\tilde{K}$ to be the $n$-skeleton of $Y$. Now the $n$-th homology of that is not necessarily torsion, but since rationalized homotopy and rational homology agree, we can choose maps from $S^n$ to $\tilde{K}$ such that they are nullhomotopic in $Y$, and their images in $H_n(\tilde{K})$ generate it rationally. But then form $K$ by attaching cells along those maps. $K$ comes with a map to $Y$, and has torsion homology.

• An equivalent argument goes like this: The Chern character from $\tilde{ko}^0(K)$ to the product of the rational cohomology groups $H^{4j}(K)$ is rationally an isomorphism if $K$ is a finite complex. Therefore in order for a vector bundle on $K$ to be such that some multiple of it is trivial it is sufficient if the rational Pontryagin classes vanish. Of course they vanish if the bundle is pulled back from a bundle on $B\Sigma_k$ (or any space having trivial rational cohomology). – Tom Goodwillie Nov 14 '15 at 14:06
• By the way, for finite $G$ the group $\tilde{ko}^0(BG)$ is indeed not a torsion group; in fact it is torsion-free, isn't it? According to the Atiyah-Segal completion theorem it is a completion of the real representation ring of $G$. – Tom Goodwillie Nov 14 '15 at 14:09
• Sorry, I'm confused, maybe you find my mistake. If you take the inclusion $\mathbb{R} P^2\rightarrow B\Sigma_2$, then the associated bundle is $E = \gamma_{\mathbb{R} P^2} \oplus \mathbb{R}$ and $w(kE) = (1+w_1)^k = 1 + kw_1 + (k-1)w_1^2$ which is never zero. – user_11437 Nov 15 '15 at 12:55
• It is $1+kw_1+\frac{k(k-1)}{2}w_1^2$, which is $1$ for $k=4n$ or $k=4n+1$. – Achim Krause Nov 15 '15 at 15:57
• @TomGoodwillie Thanks Prof. Goodwillie. In your comment, I do not understand the first two steps: "The Chern character from $\tilde{KO}^0(K)$ to the product of the rational cohomology groups $H^4j(K)$ is rationally an isomorphism if $K$ is a finite complex. Therefore in order for a vector bundle on $K$ to be such that some multiple of it is trivial it is sufficient if the rational Pontryagin classes vanish." Could you explain? Thanks! – Quan Nov 16 '15 at 5:39