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Twisted homology of symmetric groups. (English) Zbl 1003.20046
Let $$\Gamma$$ be the category of finite pointed sets and let $$T\colon\Gamma\to Ab$$ be a functor. Then $$\Sigma_n$$ acts on the pointed set $$[n]=\{0,\dots,n\}$$ and thus on $$T([n])$$. The author is interested in $$H_*(\Sigma,T)$$, defined as the colimit over $$n$$ of the $$H_*(\Sigma_n,T([n]))$$. This is analogous to the colimit over $$n$$ of the $$H_*(\text{GL}_n(k),F(k^n))$$, when $$F$$ is a functor from the category of finite dimensional vector spaces over a field $$k$$ to the category $$Ab$$. But the author emphasizes that these two cases behave rather differently. He develops theory but does not give actual examples to support this claim. He introduces ‘stable $$K$$-theory groups’ $$K^s_i(\Sigma,T)$$ and expresses $$H_*(\Sigma,T)$$ in terms of these. Then the $$K^s_i(\Sigma,\cdot)$$ themselves are shown to be derived functors of $$K^s_0(\Sigma,\cdot)$$, and they are related to cross-effects of $$T$$. For $$T$$ of finite degree (meaning that sufficiently high cross-effects vanish) and fixed $$i$$, it is shown that the $$H_i(\Sigma_n,T([n]))$$ stabilize.

MSC:
 20J05 Homological methods in group theory 18G60 Other (co)homology theories (MSC2010) 20G10 Cohomology theory for linear algebraic groups 19D55 $$K$$-theory and homology; cyclic homology and cohomology
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References:
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