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Shiquan Ren
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Let $M$ be a manifold.

Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.

Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered configuration space.

If $M$ is compact, then the graded vector space structure of $H_*(B(M,n);F)$, where $F=\mathbb{Q}$ or $\mathbb{Z}/p\mathbb{Z}$ is a field, is given in the paper C.-F. BODIGHEIMER, F. COHEN, L. TAYLOR, On the homology of configuration spaces, Topology 1989.

If $M$ is non-compact, for example, $M=S^2\times \mathbb{R}$, how to understand the graded vector space structure of $H_*(B(M,n);F)$?

Is there any reference or procedure to follow?

Let $M$ be a manifold.

Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.

Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered configuration space.

If $M$ is compact, then the graded vector space structure of $H_*(B(M,n);F)$, where $F=\mathbb{Q}$ or $\mathbb{Z}/p\mathbb{Z}$ is a field, is given in the paper C.-F. BODIGHEIMER, F. COHEN, L. TAYLOR, On the homology of configuration spaces, Topology 1989.

If $M$ is non-compact, for example, $M=S^2\times \mathbb{R}$, how to understand the graded vector space structure of $H_*(B(M,n);F)$?

Let $M$ be a manifold.

Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.

Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered configuration space.

If $M$ is compact, then the graded vector space structure of $H_*(B(M,n);F)$, where $F=\mathbb{Q}$ or $\mathbb{Z}/p\mathbb{Z}$ is a field, is given in the paper C.-F. BODIGHEIMER, F. COHEN, L. TAYLOR, On the homology of configuration spaces, Topology 1989.

If $M$ is non-compact, for example, $M=S^2\times \mathbb{R}$, how to understand the graded vector space structure of $H_*(B(M,n);F)$?

Is there any reference or procedure to follow?

Source Link
Shiquan Ren
  • 2k
  • 11
  • 22

homology of configuration spaces of non-compact manifolds

Let $M$ be a manifold.

Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.

Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered configuration space.

If $M$ is compact, then the graded vector space structure of $H_*(B(M,n);F)$, where $F=\mathbb{Q}$ or $\mathbb{Z}/p\mathbb{Z}$ is a field, is given in the paper C.-F. BODIGHEIMER, F. COHEN, L. TAYLOR, On the homology of configuration spaces, Topology 1989.

If $M$ is non-compact, for example, $M=S^2\times \mathbb{R}$, how to understand the graded vector space structure of $H_*(B(M,n);F)$?