On the higher Whitehead groups of a Bieberbach group

Nicas, Andrew J.

doi:10.1090/S0002-9947-1985-0768746-X

On the higher Whitehead groups of a Bieberbach group
HTML articles powered by AMS MathViewer

by Andrew J. Nicas PDF

Trans. Amer. Math. Soc. 287 (1985), 853-859 Request permission

Abstract:

Let $\Gamma$ be a Bieberbach group, i.e. the fundamental group of a compact flat Riemannian manifold. In this paper we show that if $p > 2$ is a prime, then the $p$-torsion subgroup of ${\text {Wh}_i}(\Gamma )$ vanishes for $0 \leq i \leq 2p - 2$, where ${\text {Wh}_i}(\Gamma )$ is the $i$th higher Whitehead group of $\Gamma$. The proof involves Farrell and Hsiang’s structure theorem for Bieberbach groups, parametrized surgery, pseudoisotopy, and Waldhausen’s algebraic $K$-theory of spaces.

References

Dan Burghelea and Richard Lashof, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 (1974), 1–36; ibid. 196 (1974), 37–50. MR 356103, DOI 10.1090/S0002-9947-1974-0356103-2
D. Burghelea and R. Lashof, Stability of concordances and the suspension homomorphism, Ann. of Math. (2) 105 (1977), no. 3, 449–472. MR 438365, DOI 10.2307/1970919
Dan Burghelea, Richard Lashof, and Melvin Rothenberg, Groups of automorphisms of manifolds, Lecture Notes in Mathematics, Vol. 473, Springer-Verlag, Berlin-New York, 1975. With an appendix (“The topological category”) by E. Pedersen. MR 0380841
David Epstein and Michael Shub, Expanding endomorphisms of flat manifolds, Topology 7 (1968), 139–141. MR 227996, DOI 10.1016/0040-9383(68)90022-0
F. T. Farrell and W. C. Hsiang, The topological-Euclidean space form problem, Invent. Math. 45 (1978), no. 2, 181–192. MR 482771, DOI 10.1007/BF01390272
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
A. E. Hatcher, Concordance spaces, higher simple-homotopy theory, and applications, 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. 3–21. MR 520490

Stability of pseudoisotopies

On the algebraic

theory of

ring spaces

Jean-Louis Loday, $K$-théorie algébrique et représentations de groupes, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 309–377 (French). MR 447373
Andrew J. Nicas, Induction theorems for groups of homotopy manifold structures, Mem. Amer. Math. Soc. 39 (1982), no. 267, vi+108. MR 668807, DOI 10.1090/memo/0267
Andrew J. Nicas, On the higher Whitehead groups of a Bieberbach group, Trans. Amer. Math. Soc. 287 (1985), no. 2, 853–859. MR 768746, DOI 10.1090/S0002-9947-1985-0768746-X
Andrew J. Nicas, On $\textrm {Wh}_{3}$ of a Bieberbach group, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1, 55–60. MR 727080, DOI 10.1017/S0305004100061302
Friedhelm Waldhausen, Algebraic $K$-theory of topological spaces. I, 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. 35–60. MR 520492

Algebraic

theory of generalized free products

108

Algebraic

theory of spaces-a manifold approach