Embeddings in generalized manifolds

Authors:
J. L. Bryant and W. Mio

Journal:
Trans. Amer. Math. Soc. **352** (2000), 1131-1147

MSC (2000):
Primary 57N35; Secondary 57P99

DOI:
https://doi.org/10.1090/S0002-9947-99-02531-3

Published electronically:
September 17, 1999

MathSciNet review:
1694282

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a ()-connected map from a compact PL -manifold to a generalized -manifold with the disjoint disks property, , is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact and proper maps that are properly ()-connected. The techniques developed lead to a general position result for arbitrary maps , , and a Whitney trick for separating submanifolds of that have intersection number 0, analogous to the well-known results when is a manifold.

**1.**J. L. Bryant,*On embeddings of compacta in Euclidean space*, Proc. Amer. Math. Soc.**23**(1969), 46-51. MR**39:6286****2.**-,*General position theorems for generalized manifolds*, Proc. Amer. Math. Soc.**98**(1986), 667-670. MR**87m:57020****3.**J. Bryant, S. Ferry, W. Mio, and S. Weinberger,*Topology of homology manifolds*, Ann. of Math.(2)**143**(1996), 435-467 MR**97b:57017****4.**J. W. Cannon,*The recognition problem: what is a topological manifold?*, Bull. Amer. Math. Soc.**84**(1978), 832-866. MR**58:13043****5.**-,*Shrinking cell-like decompositions of manifolds in codimension three*, Ann. of Math.(2)**110**(1979), 83-112. MR**80j:57013****6.**R. J. Daverman,*Decompositions of Manifolds*, Pure and Applied Math., 124, Academic Press, Orlando, Florida, 1986. MR**88a:57001****7.**R. D. Edwards,*Dimension theory*, Lecture Notes in Mathematics, Vol. 438, Springer-Verlag, Berlin, 1975. MR**52:15477****8.**S. Ferry,*Homotoping -maps to homeomorphisms*, Amer. J. Math.**101**(1979), 567-582. MR**81f:57007a****9.**J. F. P. Hudson,*Piecewise linear embeddings*, Ann. of Math.**85**(1967), 1-31. MR**35:6149****10.**W. Hurewicz, and H. Wallman,*Dimension Theory.*Princeton Univ. Press, Princeton, New Jersey, 1941. MR**3:312b****11.**R. Kirby and L. Siebenmann,*Foundational Essays on Topological Manifolds, Smoothings, and Triangulations*, Princeton University Press, Princeton, N.J., 1977. MR**58:31082****12.**A. Nicas,*Induction Theorems for groups of manifold structure sets*, Mem. Amer. Math. Soc., vol. 39, 1982. MR**83i:57026****13.**F. Quinn,*Resolutions of homology manifolds, the topological characterization of manifolds*, Invent. Math.**72**(1983), 267-284. MR**85b:57023**; Corrigendum, MR**87g:57031****14.**-,*An obstruction to the resolution of homology manifolds*, Mich. Math. J.**34**(1987), 285-292. MR**88j:57016****15.**-,*Homotopically stratified sets*, J. Amer. Math. Soc.**1**(1988), 441-499. MR**89g:57050****16.**J. Stalling,*The embedding of homotopy types into manifolds*, Unpublished.**17.**M. A. \v{S}tan'ko,*The embedding of compacta in Euclidean space*, Mat. Sb.**83**(125) (1970), 234-255; Math. USSR-Sb.**12**(1970), 234-254. MR**42:6804****18.**M. A. \v{S}tan'ko,*Approximation of compacta in in codimension greater than two*, Mat. Sb.**90**(132) (1973), 625-636; Math. USSR-Sb.**19**(1973), 615-626. MR**49:3957****19.**C. T. C. Wall,*Surgery on Compact Manifolds*, Academic Press, New York, 1970. MR**55:4217****20.**J. Walsh,*General position properties of generalized manifolds: a primer*, Proc. Idaho State Topology Conf., 1985.

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Additional Information

**J. L. Bryant**

Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306

Email:
bryant@math.fsu.edu

**W. Mio**

Affiliation:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306

Email:
mio@math.fsu.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02531-3

Keywords:
Generalized manifolds,
embeddings

Received by editor(s):
December 4, 1997

Published electronically:
September 17, 1999

Additional Notes:
Partially supported by NSF grant DMS-9626624.

Article copyright:
© Copyright 1999
American Mathematical Society