Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Splitting theorem for homology of $ {\rm GL}(R)$


Author: Stanisław Betley
Journal: Proc. Amer. Math. Soc. 108 (1990), 297-302
MSC: Primary 20J05
DOI: https://doi.org/10.1090/S0002-9939-1990-0984782-X
MathSciNet review: 984782
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that if $ \left\{ {{M_n}} \right\}$ is a stable system of coefficients for $ {\text{G}}{{\text{l}}_n}\left( R \right)$ and $ {H_0}\left( {{\text{Gl}}\left( R \right),{\text{lim}}\left( {{M_n}} \right)} \right)$ contains $ {\mathbf{Z}}$, then for any $ j$, the group $ {H_j}\left( {{\text{Gl}}\left( R \right),{\text{lim}}\left( {{M_n}} \right)} \right)$ contains $ {H_j}\left( {{\text{Gl}}\left( R \right),Z} \right)$ as a direct summand. Now let $ {\text{Gl}}\left( {\mathbf{Z}} \right)$ act on $ M\left( {\mathbf{Z}} \right)$ (matrices over $ {\mathbf{Z}}$ ) by conjugation. Then our theorem implies that the trace map $ {\text{tr:}}M\left( {\mathbf{Z}} \right) \to {\mathbf{Z}}$ is a split epimorphism on homology.


References [Enhancements On Off] (What's this?)

  • [1] H. Bass, Algebraic $ K$-theory, Benjamin, New York, 1968. MR 0249491 (40:2736)
  • [2] S. Betley, Vanishing theorems for homology of $ G\backslash R$, J. Pure and Appl. Algebra, 58 (1989) 213-226. MR 1004602 (90j:20089)
  • [3] W. G. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. (2) 111 (1980), 239-251. MR 569072 (81b:18006)
  • [4] F. T. Farrell and W. C. Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, Proc. Sympos. Pure Math. 32 (1978), 325-337. MR 520509 (80g:57043)
  • [5] T. G. Goodwillie, On the general linear group and Hochschild homology, Ann. of Math. (2) 121 (1985), 383-407. MR 786354 (86i:18013)
  • [6] W. van der Kallen, Homology stability for linear group, Invent. Math. 60 (1980), 269-295. MR 586429 (82c:18011)
  • [7] C. Kassel, Calcul algébrique de l'homologie de certains groupes de matrices, J. Algebra 80 (1983), 235-260. MR 690716 (84m:18015)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20J05

Retrieve articles in all journals with MSC: 20J05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-0984782-X
Keywords: Homology of a group, stable system of coefficients
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society