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YCor
YCor
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Let $G$ be a compact group (may bemaybe non-Lie group). . Let $B_{G}$ denote the classifying space of $G$. If $G$ contains a circle group $\mathbb{S}^{1}$, then I think that $H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $$H^{\ast }( B_{G};\mathbb{Q} )$ is not trivial, where $H^{\ast }$ is the cohomology algebra. I don't know how exactly to prove this.

I just know this: If $G$ is a torus and $H$ is a subtorus of $G$, then $% H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \longrightarrow H^{\ast }\left( B_{H};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $$H^{\ast }( B_{G};\mathbb{Q} ) \longrightarrow H^{\ast }( B_{H};\mathbb{Q} )$ is surjective.

Let $G$ be a compact group (may be non-Lie group). . Let $B_{G}$ denote the classifying space of $G$. If $G$ contains a circle group $\mathbb{S}^{1}$, then I think that $H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is not trivial, where $H^{\ast }$ is the cohomology algebra. I don't know how exactly to prove this.

I just know this: If $G$ is a torus and $H$ is a subtorus of $G$, then $% H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \longrightarrow H^{\ast }\left( B_{H};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is surjective.

Let $G$ be a compact group (maybe non-Lie group). Let $B_{G}$ denote the classifying space of $G$. If $G$ contains a circle group $\mathbb{S}^{1}$, then I think that $H^{\ast }( B_{G};\mathbb{Q} )$ is not trivial, where $H^{\ast }$ is the cohomology algebra. I don't know how exactly to prove this.

I just know this: If $G$ is a torus and $H$ is a subtorus of $G$, then $H^{\ast }( B_{G};\mathbb{Q} ) \longrightarrow H^{\ast }( B_{H};\mathbb{Q} )$ is surjective.

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Mehmet Onat
Mehmet Onat
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A question about cohomology of the classifying spaces of compact groups

Let $G$ be a compact group (may be non-Lie group). . Let $B_{G}$ denote the classifying space of $G$. If $G$ contains a circle group $\mathbb{S}^{1}$, then I think that $H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is not trivial, where $H^{\ast }$ is the cohomology algebra. I don't know how exactly to prove this.

I just know this: If $G$ is a torus and $H$ is a subtorus of $G$, then $% H^{\ast }\left( B_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \longrightarrow H^{\ast }\left( B_{H};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is surjective.