On the equivariant $K$- and $KO$-homology of some special linear groups

DOI10.2140/AGT.2021.21.3483zbMATH Open1509.19010arXiv2004.08199MaRDI QIDQ6338897

Publication date: 17 April 2020

Abstract: We compute the equivariant

K O

-homology of the classifying space for proper actions of

e x t r m {S L}_{3} (m a t h b b Z)

and

e x t r m {G L}_{3} (m a t h b b Z)

. We also compute the Bredon homology and equivariant

K

-homology of the classifying spaces for proper actions of

e x t r m {P S L}_{2} (m a t h b b Z [f r a c 1 p])

and

e x t r m {S L}_{2} (m a t h b b Z [f r a c 1 p])

for each prime

p

. Finally, we prove the Unstable Gromov-Lawson-Rosenberg Conjecture for a large class of groups whose maximal finite subgroups are odd order and have periodic cohomology.

Mathematics Subject Classification ID

Equivariant homology and cohomology in algebraic topology (55N91) (K)-theory and operator algebras (19K99) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) (57R15) Connections of group theory with homological algebra and category theory (20J99) Equivariant (K)-theory (19L47)

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