Two conjectures in the theory of Poincaré duality groups

Johnson, F. E. A.

doi:10.1090/S0002-9939-1981-0597638-6

Two conjectures in the theory of Poincaré duality groups
HTML articles powered by AMS MathViewer

by F. E. A. Johnson PDF

Proc. Amer. Math. Soc. 81 (1981), 353-354 Request permission

Abstract:

We show that it is not possible both to realise every Poincaré Duality group as an aspherical manifold and to construct, for each Poincaré complex $X$, a Poincaré Duality group having the same integral homology as $X$.

References

M. F. Atiyah, Thom complexes, Proc. London Math. Soc. (3) 11 (1961), 291–310. MR 131880, DOI 10.1112/plms/s3-11.1.291
J. M. Boardman and R. M. Vogt, Homotopy-everything $H$-spaces, Bull. Amer. Math. Soc. 74 (1968), 1117–1122. MR 236922, DOI 10.1090/S0002-9904-1968-12070-1
William Browder, Poincaré spaces, their normal fibrations and surgery, Invent. Math. 17 (1972), 191–202. MR 326743, DOI 10.1007/BF01425447
Samuel Gitler and James D. Stasheff, The first exotic class of $BF$, Topology 4 (1965), 257–266. MR 180985, DOI 10.1016/0040-9383(65)90010-8
D. M. Kan and W. P. Thurston, Every connected space has the homology of a $K(\pi ,1)$, Topology 15 (1976), no. 3, 253–258. MR 413089, DOI 10.1016/0040-9383(76)90040-9
Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 0645390
Michael Spivak, Spaces satisfying Poincaré duality, Topology 6 (1967), 77–101. MR 214071, DOI 10.1016/0040-9383(67)90016-X