# at.algebraic topology – What is equivariant chains on a illustration sphere? Answer

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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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