Connective K-Theory and the Borsuk–Ulam Theorem

Abstract

Let $k\ge 0$ and $n,\, r\ge 1$ be natural numbers, and let $\zeta = \mathrm{e}^{\pi \mathrm{i}/2^k}$. Suppose that $f : S({\mathbb C}^n) \rightarrow {\mathbb R}^{2r}$ is a continuous map on the unit sphere in ${\mathbb C}^n$ such that, for each $v\in S({\mathbb C}^n)$, $f(\zeta v)= - f(v)$. A connective K-theory Borsuk–Ulam theorem is used to show that, if $n> 2^kr$, then the covering dimension of the space of vectors $v\in S({\mathbb C}^n)$ such that $f(v)=0$ is at least $2(n-2^kr-1)$. It is shown, further, that there exists such a map f for which this zero-set has covering dimension equal to $2(n-2^kr-1) + 2^{k+2}k+1$.

Appendix

Proof of Proposition 2.9. Embedding Y as a retract of an open subspace $W\subseteq E$ of some Euclidean space E by $i: Y\hookrightarrow W$, $r : W\rightarrow Y$, we treat Y as a subspace of W. There is some $\epsilon >0$ such that the open ball $B_\epsilon (y)$ of radius $\epsilon$ is contained in W for all $y\in Y$.

By the hypothesis on the covering dimension, there exist a finite open cover $(U_i)_{i\in I}$ of Z and points $x_i\in U_i$ such that $f(U_i) \subseteq B_{\epsilon }(f(x_i))$ for all $i\in I$ and $\#\{ i\in I \mid x\in U_i\} \le d+1$ for each $x\in Z$.

Then we may choose a partition of unity $(\varphi _i)_{i\in I}$ subordinate to the cover and define C to be the simplicial complex

$$ \{ (s_i) \in {\mathbb R}^I \mid 0\le s_i\le 1,\, \sum s_i =1,\, \bigcap _{i : s_i>0} U_i \not =\emptyset \} $$

of dimension at most d.

Given $x\in Z$, let $J=\{ i\in I \mid x\in U_i\}$. Then $f(x) \in B_{\epsilon }(f(x_i))$ for all $i\in J$ and $\varphi _i(x)=0$ if $i\notin J$. So $f(x_i)\in B_\epsilon (f(x))$ for all $i\in J$. Since the ball is convex, for $0\le t\le 1$, we have $(1-t)f(x) + t\sum _i \varphi _i(x)f(x_i)\in B_\epsilon (f(x)) \subseteq W$.

So we may define a homotopy $f_t : Z\rightarrow Y$, $0\le t\le 1$, by

$$ f_t(x) = r((1-t)f(x) + t\sum _i \varphi _i(x)f(x_i)). $$

Now $f_0=f$ and $f_1$ given by $f_1(x) =r(\sum _i \varphi _i(x)f(x_i))$ factors through the map

$$ Z \rightarrow C \quad :\quad x\mapsto (\varphi _i(x)) $$

and

$$ C\rightarrow Y\quad : \quad (s_i)\mapsto r(\sum _i s_if(x_i)). $$

Indeed, given $(s_i)$ we can choose $x\in \bigcap _{i : s_i>0} U_i$, and then $f(x_i) \in B_\epsilon (f(x))$ if $s_i >0$. Thus we have constructed the required factorization up to homotopy.   $\square$

Proof of Proposition 2.11. Suppose that $\mathcal {U}$ is an open cover of $\tilde{Z}$. It is straightforward to check that, for each point $x\in \tilde{Z}$, with stabilizer $G_x\le G$, there is an open $G_x$-invariant neighbourhood $U_x$ of x such that (i) for each $g\in G$ the translate $gU_x$ is contained in some set of the cover $\mathcal {U}$ and (ii) for distinct cosets $gG_x,\, hG_x\in G/G_x$, the translates $gU_x$ and $hU_x$ are disjoint.

Write $\pi : \tilde{Z} \rightarrow Z$ for the projection. Then the open sets $\pi (U_x)$, ($x\in \tilde{Z}$), cover Z. By assumption, there is a finite open refinement $\mathcal {V}$ of the cover $(\pi (U_x))$ such that each point of Z lies in at most $d+1$ sets in $\mathcal {V}$.

For each $V\in \mathcal {V}$, choose an element $x_V\in \tilde{Z}$ such that $V\subseteq \pi (U_{x_V})$. The subsets $\pi ^{-1}(V)\cap gU_{x_V}$, $V\in \mathcal {V}$, $gG_x\in G/G_x$, constitute a finite open refinement of $\mathcal {U}$ with the property that each point of $\tilde{Z}$ lies in at most $d+1$ sets of the cover.   $\square$