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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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