A Borsuk-Ulman type theorem for $\mathbb{Z}_{4}$-actions

Abstract

Let $n=2k+1$ and the sphere $S^n$ be represented as

$S^n=\{ z=(z_1,..,z_{k+1}) \in C^{k+1}||z||=1 \}$

Consider the canonical action of the group $Z_4=\{ 1,i,-1,-i\}$ in $S^n$ defined by multiplication. The main result in the article is the following Borsuk-Ulam type theorem:

For any continuous function $f:S^n \rightarrow R^1$ consider the set

$A(f)=\{z \in S^n|f(z)=f(iz)=f(-z)=f(-iz)\}$

Then dim$A(f) \ geq n-3$.

The main corollary: For any continuous function $f:S^3 \rightarrow R^1$ there exists $z \in S^3$ such that

$$f(z)=f(iz)=f(-z)=f(-iz)$$