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at.algebraic topology – What is equivariant chains on a illustration sphere?
For a finite group $G$ and a finite-dimensional actual illustration $V$ of $G$, denote by $S^V$ the one-point compactification of $V$, with basepoint at infinity.
What is the decreased train complicated $C_*(S^V,infty)$ as an objective of the derived class of $G$-representations?
Admittedly it is a considerably open-ended query. One might account $C_*(S^V,infty)$ itself already as an “explicit” train complicated of $G$-representations. One can too write down a “more explicit” presentation of it when it comes to the lattice of subgroups of $G$ and the subrepresentations of $V$ that they pickle. Is there a pleasant succinct respond right here, maybe figuring out $C_*(S^V,infty)$ with one other identified objective within the derived class $G$-representations which might breathe unaffected from different (e.g. purely illustration theoretic) factors of perceive?
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