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Mehmet Onat
Mehmet Onat
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Oliver claims in the article (A proof of the Conner conjecture) that any action of a torus on a paracompact $ %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-acyclic space of finite cohomological dimension with finitely many orbit types has $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-acyclic fixed-point set: This follows from next theorem. This theorem is taken from W. Y. Hsiang's book, Cohomology Theory of Topological Transformation Groups (page 40)

Theorem: Let $G$ be a compact Lie group and $X$ be a $G$-space, paracompact and with finite cohomological dimension. Let $S\subset H^{\ast }\left( B_{G}\right) $ be a multiplicative system and $s\in S$. Then the localized restriction homomorphism

$ S^{-1}H_{G}^{\ast }\left( X\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( X^{s}\right) $

is an isomorphism.

If $X$ consists of only finite orbit types, then

$ S^{-1}H_{G}^{\ast }\left( X\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( X^{S}\right) $

is also an isomorphism.

But I have not seen any evidence to show that this conclusion can be obtained from this theorem. Probably Oliver means the following proposition derived from the theorem above in the same book (page 45)

Let $G=T^{r}$ (resp. $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}$) and $k=% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion $ (resp. $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}$). Then it is well known that \begin{eqnarray*} H^{\ast }\left( B_{G};k\right) &=&k\left[ t_{1},t_{2},\cdots ,t_{r}\right] \text{, }\deg t_{j}=2\text{ (resp. }1\text{) when }G=T^{r}\text{ (resp. }% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2}^{r}\text{) }k=% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \text{ (resp. }% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2}\text{)} \\ H^{\ast }\left( B_{G};k\right) &=&k\left[ t_{1},t_{2},\cdots ,t_{r}\right] \otimes \Lambda \left[ v_{1},v_{2},\cdots ,v_{r}\right] \text{, }\deg v_{j}=1% \text{, }t_{j}=\beta \left( v_{j}\right) \text{ when }G=% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}\text{) }k=% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}\text{ and }p\neq %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2} \end{eqnarray*}

In both cases, we shall denote the polynomial part by $R$, i.e. $R=k\left[ t_{1},t_{2},\cdots ,t_{r}\right] $ and set $S=R-\left\{ 0\right\} $ which is clearly a multiplicative systen lying in the center of $H^{\ast }\left( B_{G};k\right) $.

Proposition: (A. Borel) Let $G=T^{r}$ or $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}$, $X$ be a paracompact $G$-space with finite cohomological dimension, and $F=F\left( G,X\right) $ be the fixed point set. Then the following localized restriction homomorphism

$ S^{-1}H_{G}^{\ast }\left( X;k\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( F;k\right) =H^{\ast }\left( F;k\right) \otimes _{k}S^{-1}H^{\ast }\left( B_{G};k\right) $

is an isomorphism.

I think it is not very clear that Oliver's claim follows from this proposition. I don't think the claim is true for the coefficient $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $.

Oliver claims in the article (A proof of the Conner conjecture) that any action of a torus on a paracompact $ %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-acyclic space of finite cohomological dimension with finitely many orbit types has $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-acyclic fixed-point set: This follows from next theorem. This theorem is taken from W. Y. Hsiang's book, Cohomology Theory of Topological Transformation Groups (page 40)

Theorem: Let $G$ be a compact Lie group and $X$ be a $G$-space, paracompact and with finite cohomological dimension. Let $S\subset H^{\ast }\left( B_{G}\right) $ be a multiplicative system and $s\in S$. Then the localized restriction homomorphism

$ S^{-1}H_{G}^{\ast }\left( X\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( X^{s}\right) $

is an isomorphism.

If $X$ consists of only finite orbit types, then

$ S^{-1}H_{G}^{\ast }\left( X\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( X^{S}\right) $

is also an isomorphism.

But I have not seen any evidence to show that this conclusion can be obtained from this theorem. Probably Oliver means the following proposition derived from the theorem above in the same book (page 45)

Let $G=T^{r}$ (resp. $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}$) and $k=% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion $ (resp. $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}$). Then it is well known that \begin{eqnarray*} H^{\ast }\left( B_{G};k\right) &=&k\left[ t_{1},t_{2},\cdots ,t_{r}\right] \text{, }\deg t_{j}=2\text{ (resp. }1\text{) when }G=T^{r}\text{ (resp. }% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2}^{r}\text{) }k=% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \text{ (resp. }% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2}\text{)} \\ H^{\ast }\left( B_{G};k\right) &=&k\left[ t_{1},t_{2},\cdots ,t_{r}\right] \otimes \Lambda \left[ v_{1},v_{2},\cdots ,v_{r}\right] \text{, }\deg v_{j}=1% \text{, }t_{j}=\beta \left( v_{j}\right) \text{ when }G=% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}\text{) }k=% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}\text{ and }p\neq %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2} \end{eqnarray*}

In both cases, we shall denote the polynomial part by $R$, i.e. $R=k\left[ t_{1},t_{2},\cdots ,t_{r}\right] $ and set $S=R-\left\{ 0\right\} $ which is clearly a multiplicative systen lying in the center of $H^{\ast }\left( B_{G};k\right) $.

Proposition: (A. Borel) Let $G=T^{r}$ or $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}$, $X$ be a paracompact $G$-space with finite cohomological dimension, and $F=F\left( G,X\right) $ be the fixed point set. Then the following localized restriction homomorphism

$ S^{-1}H_{G}^{\ast }\left( X;k\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( F;k\right) =H^{\ast }\left( F;k\right) \otimes _{k}S^{-1}H^{\ast }\left( B_{G};k\right) $

is an isomorphism.

I think it is not very clear that Oliver's claim follows from this proposition. I don't think the claim is true for the coefficient $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $.

Oliver claims in the article (A proof of the Conner conjecture) that any action of a torus on a paracompact $ %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-acyclic space of finite cohomological dimension with finitely many orbit types has $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-acyclic fixed-point set: This follows from next theorem. This theorem is taken from W. Y. Hsiang's book, Cohomology Theory of Topological Transformation Groups (page 40)

Theorem: Let $G$ be a compact Lie group and $X$ be a $G$-space, paracompact and with finite cohomological dimension. Let $S\subset H^{\ast }\left( B_{G}\right) $ be a multiplicative system and $s\in S$. Then the localized restriction homomorphism

$ S^{-1}H_{G}^{\ast }\left( X\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( X^{s}\right) $

is an isomorphism.

If $X$ consists of only finite orbit types, then

$ S^{-1}H_{G}^{\ast }\left( X\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( X^{S}\right) $

is also an isomorphism.

But I have not seen any evidence to show that this conclusion can be obtained from this theorem. Probably Oliver means the following proposition derived from the theorem above in the same book (page 45)

Let $G=T^{r}$ (resp. $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}$) and $k=% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion $ (resp. $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}$). Then it is well known that \begin{eqnarray*} H^{\ast }\left( B_{G};k\right) &=&k\left[ t_{1},t_{2},\cdots ,t_{r}\right] \text{, }\deg t_{j}=2\text{ (resp. }1\text{) when }G=T^{r}\text{ (resp. }% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2}^{r}\text{) }k=% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \text{ (resp. }% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2}\text{)} \\ H^{\ast }\left( B_{G};k\right) &=&k\left[ t_{1},t_{2},\cdots ,t_{r}\right] \otimes \Lambda \left[ v_{1},v_{2},\cdots ,v_{r}\right] \text{, }\deg v_{j}=1% \text{, }t_{j}=\beta \left( v_{j}\right) \text{ when }G=% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}\text{) }k=% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}\text{ and }p\neq %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2} \end{eqnarray*}

In both cases, we shall denote the polynomial part by $R$, i.e. $R=k\left[ t_{1},t_{2},\cdots ,t_{r}\right] $ and set $S=R-\left\{ 0\right\} $ which is clearly a multiplicative systen lying in the center of $H^{\ast }\left( B_{G};k\right) $.

Proposition: (A. Borel) Let $G=T^{r}$ or $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}$, $X$ be a paracompact $G$-space with finite cohomological dimension, and $F=F\left( G,X\right) $ be the fixed point set. Then the following localized restriction homomorphism

$ S^{-1}H_{G}^{\ast }\left( X;k\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( F;k\right) =H^{\ast }\left( F;k\right) \otimes _{k}S^{-1}H^{\ast }\left( B_{G};k\right) $

is an isomorphism.

I don't think the claim is true for the coefficient $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $.

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Mehmet Onat
Mehmet Onat
  • 1.4k
  • 7
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A question about acyclic fixed point set of torus action on acyclic space

Oliver claims in the article (A proof of the Conner conjecture) that any action of a torus on a paracompact $ %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-acyclic space of finite cohomological dimension with finitely many orbit types has $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-acyclic fixed-point set: This follows from next theorem. This theorem is taken from W. Y. Hsiang's book, Cohomology Theory of Topological Transformation Groups (page 40)

Theorem: Let $G$ be a compact Lie group and $X$ be a $G$-space, paracompact and with finite cohomological dimension. Let $S\subset H^{\ast }\left( B_{G}\right) $ be a multiplicative system and $s\in S$. Then the localized restriction homomorphism

$ S^{-1}H_{G}^{\ast }\left( X\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( X^{s}\right) $

is an isomorphism.

If $X$ consists of only finite orbit types, then

$ S^{-1}H_{G}^{\ast }\left( X\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( X^{S}\right) $

is also an isomorphism.

But I have not seen any evidence to show that this conclusion can be obtained from this theorem. Probably Oliver means the following proposition derived from the theorem above in the same book (page 45)

Let $G=T^{r}$ (resp. $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}$) and $k=% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion $ (resp. $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}$). Then it is well known that \begin{eqnarray*} H^{\ast }\left( B_{G};k\right) &=&k\left[ t_{1},t_{2},\cdots ,t_{r}\right] \text{, }\deg t_{j}=2\text{ (resp. }1\text{) when }G=T^{r}\text{ (resp. }% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2}^{r}\text{) }k=% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \text{ (resp. }% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2}\text{)} \\ H^{\ast }\left( B_{G};k\right) &=&k\left[ t_{1},t_{2},\cdots ,t_{r}\right] \otimes \Lambda \left[ v_{1},v_{2},\cdots ,v_{r}\right] \text{, }\deg v_{j}=1% \text{, }t_{j}=\beta \left( v_{j}\right) \text{ when }G=% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}\text{) }k=% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}\text{ and }p\neq %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{2} \end{eqnarray*}

In both cases, we shall denote the polynomial part by $R$, i.e. $R=k\left[ t_{1},t_{2},\cdots ,t_{r}\right] $ and set $S=R-\left\{ 0\right\} $ which is clearly a multiplicative systen lying in the center of $H^{\ast }\left( B_{G};k\right) $.

Proposition: (A. Borel) Let $G=T^{r}$ or $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}^{r}$, $X$ be a paracompact $G$-space with finite cohomological dimension, and $F=F\left( G,X\right) $ be the fixed point set. Then the following localized restriction homomorphism

$ S^{-1}H_{G}^{\ast }\left( X;k\right) \longrightarrow S^{-1}H_{G}^{\ast }\left( F;k\right) =H^{\ast }\left( F;k\right) \otimes _{k}S^{-1}H^{\ast }\left( B_{G};k\right) $

is an isomorphism.

I think it is not very clear that Oliver's claim follows from this proposition. I don't think the claim is true for the coefficient $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $.