The complete stable splitting for the classifying space of a finite group
This paper describes the complete stable splittings of the classifying space \(BG\) for all finite groups \(G\). It is enough to be able to compute the stable homotopy classes \(\{BG,BG\}\) for each Sylow \(p\)-subgroup \(P\) of \(G\). This is known from Carlsson's proof of the Segal Conjecture to be isomorphic to the \(p\)-adic completion of the double Burnside ring \(A(P,P)\). There is an ideal \(J_ p\subseteq \{BP,BP\}\) generated by maps of the form \(BP\to BQ\to BP\) where \(Q\subset P\) and for this ideal \(\hat Z_ p\text{Out}(P)\approx \{BP,BP\}/J_ p\). If \(X\) is a summand of \(\{BG,BG\}\) then it corresponds to a primitive idempotent \(e\in\{BP,BP\}\). If \(e\) is not in \(J_ p\) it is called a dominant summand. The multiplicity of such a summand \(X\) is equal to the dimension of \(\mathbb{F}_ p\text{Out}(P)\) over its endomorphism field. By calculating the dimension of these modules the authors obtain the result on the splitting.
scientific article; zbMATH DE number 54635