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Community Bot
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Let $H$ be a finite group acting on a group $G$. Are there general conditions one can place on $G$ so that the set $H^1(H, G)$ is guaranteed to be finite? For instance, does finiteness hold if $G$ is finitely presented, or if $K(G, 1)$ is a finite CW complex?

Added: According to Reid Barton's answer herehere, $H^1(H, G)$ is identified with $\pi_0(X^{hH})$, where $X$ is the space $K(G, 1)$ and the $hH$ denotes homotopy invariants. Thus it would be enough to know the following: if a finite group $H$ acts on a finite CW complex $X$, is $\pi_0(X^{hH})$ finite?

Let $H$ be a finite group acting on a group $G$. Are there general conditions one can place on $G$ so that the set $H^1(H, G)$ is guaranteed to be finite? For instance, does finiteness hold if $G$ is finitely presented, or if $K(G, 1)$ is a finite CW complex?

Added: According to Reid Barton's answer here, $H^1(H, G)$ is identified with $\pi_0(X^{hH})$, where $X$ is the space $K(G, 1)$ and the $hH$ denotes homotopy invariants. Thus it would be enough to know the following: if a finite group $H$ acts on a finite CW complex $X$, is $\pi_0(X^{hH})$ finite?

Let $H$ be a finite group acting on a group $G$. Are there general conditions one can place on $G$ so that the set $H^1(H, G)$ is guaranteed to be finite? For instance, does finiteness hold if $G$ is finitely presented, or if $K(G, 1)$ is a finite CW complex?

Added: According to Reid Barton's answer here, $H^1(H, G)$ is identified with $\pi_0(X^{hH})$, where $X$ is the space $K(G, 1)$ and the $hH$ denotes homotopy invariants. Thus it would be enough to know the following: if a finite group $H$ acts on a finite CW complex $X$, is $\pi_0(X^{hH})$ finite?

Added rephrasing of quesiton in terms of homotopy invariants
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Winnie
Winnie
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Let $H$ be a finite group acting on a group $G$. Are there general conditions one can place on $G$ so that the set $H^1(H, G)$ is guaranteed to be finite? For instance, does finiteness hold if $G$ is finitely presented, or if $K(G, 1)$ is a finite CW complex?

Added: According to Reid Barton's answer here, $H^1(H, G)$ is identified with $\pi_0(X^{hH})$, where $X$ is the space $K(G, 1)$ and the $hH$ denotes homotopy invariants. Thus it would be enough to know the following: if a finite group $H$ acts on a finite CW complex $X$, is $\pi_0(X^{hH})$ finite?

Let $H$ be a finite group acting on a group $G$. Are there general conditions one can place on $G$ so that the set $H^1(H, G)$ is guaranteed to be finite? For instance, does finiteness hold if $G$ is finitely presented, or if $K(G, 1)$ is a finite CW complex?

Let $H$ be a finite group acting on a group $G$. Are there general conditions one can place on $G$ so that the set $H^1(H, G)$ is guaranteed to be finite? For instance, does finiteness hold if $G$ is finitely presented, or if $K(G, 1)$ is a finite CW complex?

Added: According to Reid Barton's answer here, $H^1(H, G)$ is identified with $\pi_0(X^{hH})$, where $X$ is the space $K(G, 1)$ and the $hH$ denotes homotopy invariants. Thus it would be enough to know the following: if a finite group $H$ acts on a finite CW complex $X$, is $\pi_0(X^{hH})$ finite?

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Winnie
Winnie
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Finiteness of non-abelian cohomology

Let $H$ be a finite group acting on a group $G$. Are there general conditions one can place on $G$ so that the set $H^1(H, G)$ is guaranteed to be finite? For instance, does finiteness hold if $G$ is finitely presented, or if $K(G, 1)$ is a finite CW complex?