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A simplicial approach to spaces of homomorphisms

University of British Columbia

Let G be a real linear algebraic group and L a finitely generated cosimplicial group. We prove that the space of homomorphisms Hom(Ln;G) has a homotopy stable decomposition for each n ≥ 1. When G is a compact Lie group, we show that the decomposition is G-equivariant with respect to the induced action of conjugation by elements of G. In particular, under these hypotheses on G, we obtain stable decompositions for Hom(Fqn;G) and Rep(Fqn;G) respectively, where Fqn are the finitely generated free nilpotent groups of nilpotency class q-1. The spaces Hom(Ln;G) assemble into a simplicial space Hom(L;G). When G=U we show that its geometric realization B(L;U) has a non-unital E-infinity-ring space structure whenever Hom(L0;U(m)) is path connected for all m ≥ 0. We describe the connected components of Hom(Fqn;SU(2)) arising from non-commuting q-nilpotent n-tuples. We prove this by showing that all these n-tuples are conjugated to n-tuples consisting of elements in the the generalized quaternion groups Q2q in SU(2), of order 2^q. Using this result, we exhibit the homotopy type of SHom(Fqn;SU(2)) and a homotopy description of the classifying spaces B(q;SU(2)) of transitionally q-nilpotent principal SU(2)-bundles. The above computations are also done for SO(3) and U(2). Finally, for q = 2, the space B(2;G) is denoted BcomG, and we compute the integral cohomology ring for the Lie groups G = SU(2) andU(2). We also include cohomology calculations for the spaces BcomQ2q .

Text

2017-08-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/62685

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/