Desingularizing homology manifolds

Bryant, J L¹ ; Ferry, Steven² ; Mio, Washington¹ ; Weinberger, Shmuel³

¹ Department of Mathematics, Florida State University, Tallahassee FL 32306, USA
² Department of Mathematical Sciences, Rutgers University, Piscataway NJ 08854, USA
³ Department of Mathematics, The University of Chicago, Chicago IL 60637, USA

Geometry & topology, Tome 11 (2007) no. 3, pp. 1289-1314.

Résumé

We prove that if $X^n$ , $n\ge 6$, is a compact ANR homology $n$–manifold, we can blow up the singularities of $X$ to obtain an ANR homology $n$–manifold with the disjoint disks property. More precisely, we show that there is an ANR homology $n$–manifold $Y$ with the disjoint disks property and a cell-like map $f:Y\to X$.

Détail

DOI : 10.2140/gt.2007.11.1289

Keywords: homology manifold, cell-like map, controlled topology

Bibliographie

[1] M Bestvina, Characterizing $k$–dimensional universal Menger compacta, Mem. Amer. Math. Soc. 71 (1988)

[2] M Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960) 478

[3] J L Bryant, General position theorems for generalized manifolds, Proc. Amer. Math. Soc. 98 (1986) 667

[4] J Bryant, S Ferry, W Mio, S Weinberger, Topology of homology manifolds, Ann. of Math. $(2)$ 143 (1996) 435

[5] A V Cernavskii, A generalization of Keldysh's construction of a monotone map of a cube onto a cube of higher dimension, Russian Math. Surveys 40 (1985) 165

[6] T A Chapman, Approximation results in topological manifolds, Mem. Amer. Math. Soc. 34 (1981)

[7] T A Chapman, L C Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976) 171

[8] R J Daverman, L S Husch, Decompositions and approximate fibrations, Michigan Math. J. 31 (1984) 197

[9] J Dugundji, Topology, Allyn and Bacon (1978)

[10] R D Edwards, The topology of manifolds and cell-like maps, from: "Proceedings of the International Congress of Mathematicians (Helsinki, 1978)", Acad. Sci. Fennica (1980) 111

[11] S Ferry, Epsilon-delta surgery over $\mathbb{Z}$

[12] S Ferry, A stable converse to the Vietoris–Smale theorem with applications to shape theory, Trans. Amer. Math. Soc. 261 (1980) 369

[13] S C Ferry, E K Pedersen, Epsilon surgery theory, from: "Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993)", London Math. Soc. Lecture Notes 227, Cambridge Univ. Press (1995) 167

[14] M W Hirsch, On normal microbundles, Topology 5 (1966) 229

[15] C B Hughes, Approximate fibrations on topological manifolds, Michigan Math. J. 32 (1985) 167

[16] K Kawamura, An inverse system approach to Menger manifolds, Topology Appl. 61 (1995) 281

[17] R C Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977) 495

[18] E K Pedersen, F Quinn, A Ranicki, Controlled surgery with trivial local fundamental groups, from: "High-dimensional manifold topology", World Sci. Publ., River Edge, NJ (2003) 421

[19] F Quinn, Ends of maps. I, Ann. of Math. $(2)$ 110 (1979) 275

[20] F Quinn, Resolutions of homology manifolds, and the topological characterization of manifolds, Invent. Math. 72 (1983) 267

[21] C P Rourke, B J Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer (1982)

[22] C T C Wall, Surgery on compact manifolds, London Math. Soc. Monographs 1, Academic Press (1970)