Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Homology manifold bordism

Authors: Heather Johnston and Andrew Ranicki
Journal: Trans. Amer. Math. Soc. 352 (2000), 5093-5137
MSC (2000): Primary 57P05; Secondary 19J25
Published electronically: March 16, 2000
MathSciNet review: 1778506
Full-text PDF

Abstract | References | Similar Articles | Additional Information


The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact $ANR$ homology manifolds of dimension $\geq 6$ is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.

First, we establish homology manifold transversality for submanifolds of dimension $\geq 7$: if $f:M \to P$ is a map from an $m$-dimensional homology manifold $M$ to a space $P$, and $Q \subset P$ is a subspace with a topological $q$-block bundle neighborhood, and $m-q \geq 7$, then $f$ is homology manifold $s$-cobordant to a map which is transverse to $Q$, with $f^{-1}(Q) \subset M$ an $(m-q)$-dimensional homology submanifold.

Second, we obtain a codimension $q$ splitting obstruction $s_Q(f) \in LS_{m-q}(\Phi)$ in the Wall $LS$-group for a simple homotopy equivalence $f:M \to P$ from an $m$-dimensional homology manifold $M$ to an $m$-dimensional Poincaré space $P$ with a codimension $q$ Poincaré subspace $Q \subset P$ with a topological normal bundle, such that $s_Q(f)=0$ if (and for $m-q \geq 7$ only if) $f$ splits at $Q$ up to homology manifold $s$-cobordism.

Third, we obtain the multiplicative structure of the homology manifold bordism groups $\Omega^H_*\cong\Omega^{TOP}_*[L_0(\mathbb Z)]$.

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

  • 1. J. Bryant, S. Ferry, W. Mio, S. Weinberger, Topology of homology manifolds, Ann. of Math. 143, 435-467 (1996). MR 97b:57017
  • 2. T.A. Chapman, Simple homotopy theory for $ANR$'s, General Topology and its Applications 7, 165-174 (1977). MR 58:18414
  • 3. M. Cohen, Simplicial structures and transverse cellularity, Ann. of Math. 85, 218-245 (1967). MR 35:1037
  • 4. S. Ferry, E.K. Pedersen, Epsilon surgery theory I, In: Novikov Conjectures, Index Theorems and Rigidity, Vol. 2 (Oberwolfach, 1993), Lond. Math. Soc. Lecture Note Ser. 227, Cambridge University Press (1995), 167-226. MR 97g:57044
  • 5. B. Hughes, L. Taylor, B. Williams, Manifold approximate fibrations are approximately bundles, Forum Math. 3, 309-325 (1991). MR 92k:57040
  • 6. H. Johnston, Transversality for homology manifolds, Topology 38, 673-697 (1999). MR 99k:57048
  • 7. L. Jones, Patch spaces: a geometric representation for Poincaré spaces, Ann. of Math. 97, 306-343 (1973); 102, 183-185 (1975) MR 47:4269; MR 52:11930
  • 8. R. Kirby, L. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Ann. of Math. Study 88, Princeton University Press (1977). MR 58:31082
  • 9. N. Levitt, A. Ranicki, Intrinsic transversality structures, Pacific J. Math. 129, 85-144 (1987). MR 88m:57027
  • 10. A. Marin, La transversalité topologique, Ann. of Math. 106, 269-293 (1977). MR 57:10707
  • 11. F. Quinn, Resolutions of homology manifolds, and the topological characterization of manifolds, Invent. Math. 72, 264-284 (1983); Corrigendum 85 (1986) 653. MR 85b:57023; MR 87g:57031
  • 12. -, An obstruction to the resolution of homology manifolds, Michigan Math. J. 34, 284-291 (1987). MR 88j:57016
  • 13. -, Topological transversality holds in all dimensions, Bull. Amer. Math. Soc. 18, 145-148 (1988). MR 89c:57016
  • 14. A. Ranicki, The algebraic theory of surgery II. Applications to topology, Proc. Lond. Math. Soc. 40, 193-287 (1980). MR 82f:57024b
  • 15. -, Exact sequences in the algebraic theory of surgery, Mathematical Notes 26, Princeton University Press (1981). MR 82h:57027
  • 16. -, Algebraic $L$-theory and topological manifolds, Cambridge Tracts in Mathematics 102, Cambridge University Press (1992). MR 94i:57051
  • 17. C.P. Rourke, B.J. Sanderson, On topological neighbourhoods, Compositio Math. 22, 387-425 (1970). MR 45:7720
  • 18. C.T.C. Wall, Surgery on compact manifolds, Academic Press (1970); 2nd edition, Mathematical Monographs and Surveys 69, A.M.S. (1999) CMP 99:12; MR 55:4217 (1st ed.)
  • 19. S. Weinberger, Nonlocally linear manifolds and orbifolds, Proc. 1994 Zürich ICM, 637-647, Birkhäuser (1995). MR 97g:57028
  • 20. S. Weinberger, The topological classification of stratified spaces, Chicago Lectures in Mathematics, University of Chicago Press, (1994). MR 96b:57024
  • 21. J. West, Mapping Hilbert cube manifolds to $ANR$'s, Ann. of Math. 106, 1-18 (1977). MR 56:9534

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57P05, 19J25

Retrieve articles in all journals with MSC (2000): 57P05, 19J25

Additional Information

Heather Johnston
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003

Andrew Ranicki
Affiliation: Department of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK

Keywords: Homology manifolds, bordism, transversality, surgery
Received by editor(s): March 25, 1998
Published electronically: March 16, 2000
Additional Notes: This work was carried out in connection with the first named author’s EPSRC Visiting Fellowship in Edinburgh in August, 1997.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society