Generalized -manifolds whose nonmanifold set has neighborhoods bounded by tori

Author:
Matthew G. Brin

Journal:
Trans. Amer. Math. Soc. **264** (1981), 539-555

MSC:
Primary 57P99; Secondary 57N10

DOI:
https://doi.org/10.1090/S0002-9947-1981-0603780-8

MathSciNet review:
603780

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that all compact, ANR, generalized -manifolds whose nonmanifold set is 0-dimensional and has a neighborhood system bounded by tori are cell-like images of compact -manifolds if and only if the Poincaré conjecture is true. We also discuss to what extent the assumption of the Poincaré conjecture can be replaced by other hypotheses.

**[1]**S. Armentrout,*Cellular decompositions of**-manifolds that yield**-manifolds*, Mem. Amer. Math. Soc. no. 107 (1971). MR**0413104 (54:1225)****[2]**R. H. Bing,*Approximating surfaces with polyhedral ones*, Ann. of Math. (2)**65**(1957), 456-483. MR**0087090 (19:300f)****[3]**-,*Necessary and sufficient conditions that a**-manifold be*, Ann. of Math. (2)**68**(1958), 17-37. MR**0095471 (20:1973)****[4]**K. Borsuk,*Theory of retracts*, Monografie Mat., Tom. 44, PWN, Warsaw, 1967. MR**0216473 (35:7306)****[5]**M. G. Brin,*Improving**-manifold compactifications of open**-manifolds*, Houston J. Math.**4**(1978), 149-163. MR**0494116 (58:13046)****[6]**J. W. Cannon,*The recognition problem: What is a topological manifold?*, Bull. Amer. Math. Soc.**84**(1978), 832-867. MR**0494113 (58:13043)****[7]**J. W. Cannon, J. L. Bryant and R. C. Lacher,*The structure of generalized manifolds having nonmanifold set of trivial dimension*, Geometric Topology (Proc. 1977 Georgia Topology Conf.), edited by J. C. Cantrell, Academic Press, New York, 1979, pp. 261-303. MR**537735 (80h:57026)****[8]**C. H. Edwards, Jr.,*Open**-manifolds which are simply connected at infinity*, Proc. Amer. Math. Soc.**14**(1963), 391-395. MR**0150745 (27:732)****[9]**W. Haken,*Some results on surfaces in**-manifolds*, Studies in Modern Topology, Math. Assoc. Amer., distributed by Prentice-Hall, Englewood Cliffs, N. J., 1968, pp. 39-98. MR**0224071 (36:7118)****[10]**J. M. Kister and D. R. McMillan, Jr.,*Locally Euclidean factors of**which cannot be embedded in*, Ann. of Math. (2)**76**(1962), 541-546. MR**0144322 (26:1868)****[11]**T. Knoblauch,*Imbedding deleted**-manifold neighborhoods in*, Illinois J. Math.**18**(1974), 598-601. MR**0380801 (52:1698)****[12]**R. C. Lacher,*Cell-like mappings*. I, Pacific J. Math.**30**(1969), 717-731. MR**0251714 (40:4941)****[13]**-,*Cell-like mappings and their generalizations*, Bull. Amer. Math. Soc.**83**(1977), 495-553. MR**0645403 (58:31095)****[14]**D. R. McMillan, Jr.,*Some contractible open**-manifolds*, Trans. Amer. Math. Soc.**102**(1962), 373-382. MR**0137105 (25:561)****[15]**-,*Strong homotopy equivalence of**-manifolds*, Bull. Amer. Math. Soc.**73**(1967), 718-722. MR**0229243 (37:4817)****[16]**-,*Acyclicity in three-manifolds*, Bull. Amer. Math. Soc.**76**(1970), 942-964. MR**0270380 (42:5269)****[17]**C. D. Papakyriakopoulos,*On Dehn's lemma and the asphericity of knots*, Ann. of Math. (2)**66**(1957), 1-26. MR**0090053 (19:761a)****[18]**F. Quinn,*Resolutions of homology manifolds*, Notices Amer. Math. Soc.**26**(1979), A-130, Abstract #763-57-9.**[19]**F. Raymond,*R. L. Wilder's work on generalized manifolds--an appreciation*, Algebraic and Geometric Topology (K. C. Millett, Ed.), Lecture Notes in Math., vol. 664, Springer-Verlag, Berlin and New York, 1978, pp. 7-32. MR**518402 (80h:01023)****[20]**R. Sternfeld,*A contractible open**-manifold that embeds in no compact**-manifold*, Ph.D. Thesis, Univ. Wisconsin, Madison, 1977.**[21]**J. Vrabec,*Submanifolds of acyclic**-manifolds*, Pacific J. Math.**49**(1973), 243-263. MR**0350742 (50:3234)****[22]**F. Waldhausen,*Irreducible**-manifolds which are sufficiently large*, Ann. of Math. (2)**87**(1968), 56-88. MR**0224099 (36:7146)****[23]**A. Wright and R. Messer,*Embedding open**-manifolds in compact**-manifolds*, Pacific J. Math.**82**(1979), 163-177. MR**549841 (80m:57004)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57P99,
57N10

Retrieve articles in all journals with MSC: 57P99, 57N10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0603780-8

Article copyright:
© Copyright 1981
American Mathematical Society