### On nonsymmetric theorems for $(H,G)$-coincidences

DOI: http://dx.doi.org/10.12775/TMNA.2009.008

#### Abstract

Let $X$ be a compact Hausdorff space, $\varphi\colon X\to

S^{n}$ a continuous map into the $n$-sphere $S^n$ that induces a

nonzero homomorphism $\varphi^{*}\colon H^{n}(S^{n};{\mathbb Z}_{p})\to

H^{n}(X;{\mathbb Z}_{p})$, $Y$ a $k$-dimensional CW-complex and

$f\colon X\to Y$ a continuous map. Let $G$ a finite group which acts

freely on $S^{n}$. Suppose that $H\subset G$ is a normal cyclic

subgroup of a prime order. In this paper, we define and we estimate

the cohomological dimension of the set $A_{\varphi}(f,H,G)$ of

$(H,G)$-coincidence points of $f$ relative to $\varphi$.

S^{n}$ a continuous map into the $n$-sphere $S^n$ that induces a

nonzero homomorphism $\varphi^{*}\colon H^{n}(S^{n};{\mathbb Z}_{p})\to

H^{n}(X;{\mathbb Z}_{p})$, $Y$ a $k$-dimensional CW-complex and

$f\colon X\to Y$ a continuous map. Let $G$ a finite group which acts

freely on $S^{n}$. Suppose that $H\subset G$ is a normal cyclic

subgroup of a prime order. In this paper, we define and we estimate

the cohomological dimension of the set $A_{\varphi}(f,H,G)$ of

$(H,G)$-coincidence points of $f$ relative to $\varphi$.

#### Keywords

Borsuk-Ulam theorem; ${\Bbb Z}_{p}$-index; $(H;G)$-coincidence; free actions

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