Homology of linear groups via cycles in $BG\times X$

Kevin P. Knudson; Mark E. Walker

paper

Homology of linear groups via cycles in $B G \times X$

arXiv:math/0311362

Abstract

Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups $H_{i} (X, G)$ by considering cycles in the simplicial scheme $B G \times X (ani d e a s ug g es t e d b y A n d r e i S u s l in) . W e d i sc u ss t h e ba s i c p r o p er t i eso f t h ese g r o u p s an d co n s t r u c t a s p ec t r a l se q u e n ce, b e g innin g w i t h t h e g r o u p s$ H_i(Δ^j,G) $, w hi c h co n v er g es t o t h ee t a l eco h o m o l o g y o f t h es im pl i c ia l g r o u pB G . T h ese g r o u p s a r e t h er e f or eco nn ec t e d w i t h t h es t u d y o f F r i e d l an d e r^{'} s g e n er a l i z e d i so m or p hi s m co nj ec t u r e . < p > W e a l soco m p u t eso m ee x am pl es, f oc u s in g in p a r t i c u l a r o n t h ec a se X = S p ec (k) . I n t h ec a se w h er e k i s t h er e a l n u mb er s, t h er e i s a co nn ec t i o nb e tw ee n t h e g r o u p s$ H_i $an d t h e Z /2 - e q u i v a r ian t co h o m o l o g y o f t h ec l a ss i f y in g s p a ceo f t h e d i scr e t e g r o u p$ G(\mathbb R)$.

15 pages

References in corpus (1)

An Atiyah-Hirzebruch spectral sequence for KR-theory

Homology of linear groups via cycles in BG×X

Abstract

References in corpus (1)

Homology of linear groups via cycles in $B G \times X$