Let $G$ be a compact group. Then there exists a universal principal $G$-bundle $G\rightarrow E_{G}\rightarrow B_{G}$. Let $X$ be a paracompact $G$-space and suppose that $X\rightarrow X_{G}\rightarrow B_{G}$ is fiber bundle (called as the Borel fibration) associated to the universal bundle. Its structure group is $G$.

Theorem (The cohomology Leray-Serre spectral sequence]) For the Borel fibration $X\overset{i}{\rightarrow }X_{G}\rightarrow B_{G}$, there is a first quadrant spectral sequence of algebras $\left\{ E_{r}^{\ast ,\ast },d_{r}\right\} $, with $$ E_{2}^{p,q}\cong H^{p}\left( B_{G};\mathcal{H}^{q}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) \right) $$ the cohomology of $B_{G}$ with local coefficients in the cohomology of $X$, and coverging to $H^{\ast }\left( X_{G};% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ as an algebra.

Under some conditions, the local coefficients system is simple, that is, $% \mathcal{H}^{q}\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $ is the usual coefficient $H^{\ast }\left( X;% %TCIMACRO{\U{211a} }% %BeginExpansion \mathbb{Q} %EndExpansion \right) $. For example, the local coefficients system is simple if $B_{G}$ is simply connected (for example, the case $G$ is compact connected Lie group)

My Question: Suppose that $G$ is any compact connected group (not Lie group), and $X$ is a path connected $G$-space, then local coefficient system is simple?

Or under which conditions, the local coefficients system is simple.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.