Splitting theorem for homology of 𝐺𝐿(𝑅)

Betley, Stanisław

doi:10.1090/S0002-9939-1990-0984782-X

Splitting theorem for homology of $\textrm {GL}(R)$
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by Stanisław Betley PDF

Proc. Amer. Math. Soc. 108 (1990), 297-302 Request permission

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

Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Stanisław Betley, Vanishing theorems for homology of $\textrm {Gl}_nR$, J. Pure Appl. Algebra 58 (1989), no. 3, 213–226. MR 1004602, DOI 10.1016/0022-4049(89)90037-6
W. G. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. (2) 111 (1980), no. 2, 239–251. MR 569072, DOI 10.2307/1971200
F. T. Farrell and W. C. Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 325–337. MR 520509
Thomas G. Goodwillie, On the general linear group and Hochschild homology, Ann. of Math. (2) 121 (1985), no. 2, 383–407. MR 786354, DOI 10.2307/1971179
Wilberd van der Kallen, Homology stability for linear groups, Invent. Math. 60 (1980), no. 3, 269–295. MR 586429, DOI 10.1007/BF01390018
Christian Kassel, Calcul algébrique de l’homologie de certains groupes de matrices, J. Algebra 80 (1983), no. 1, 235–260 (French). MR 690716, DOI 10.1016/0021-8693(83)90030-3