Algebraic Topology and Related Topics pp 51-66 | Cite as

# 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 \(2(n-2^kr-1) + 2^{k+2}k+1\).

## Keywords

Borsuk–Ulam theorem Connective*K*-theory

*K*-theory Euler class

## 2010 Mathematics Subject Classification

Primary: 55M25 55N15 55R25 Secondary: 55R40 55R70 55R91## Notes

### Acknowledgements

I am grateful to Prof. Mahender Singh for discussions on some of the material in Sect. 5.

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