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A complement theorem for shape concordant compacta. (English) Zbl 0553.57006

Throughout the following suppose X and Y are compacta of polyhedral shape contained in a piecewise linear (PL) manifold M of dimension \(n\geq 6\) and with empty boundary. The author sharpens the notion of ”shape concordance” [cf. L. S. Husch and I. Ivanšic, Lect. Notes Math. 870, 135-149 (1981; Zbl 0473.57004)] to define ”ILC shape concordance” and proves the following theorem: If X and Y are ILC shape concordant in M and X has the shape of a polyhedron of dimension at most (n-3), then the complements (M-X) and (M-Y) are homeomorphic. Roughly, the theorem states, ”ILC concordance implies homeomorphic complements.” This is analogous to J. F. P. Hudson’s, ”concordance implies isotopy” for codimension 3 embeddings of polyhedra in PL manifolds [cf. Ann. Math., II. Ser. 91, 425-448 (1970; Zbl 0202.546)]. For additional results on ”complement theorems in shape theory” see Chapter III in Shape theory and geometric topology, Proc. Conf., Dubrovnik 1981, Lect. Notes Math. 870 (1981; Zbl 0456.00028).
Reviewer: S.Singh

MSC:

57N25 Shapes (aspects of topological manifolds)
57N70 Cobordism and concordance in topological manifolds
54C56 Shape theory in general topology
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[1] Karol Borsuk, Theory of shape, PWN — Polish Scientific Publishers, Warsaw, 1975. Monografie Matematyczne, Tom 59. · Zbl 0317.55006
[2] J. F. P. Hudson, Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0189.54507
[3] J. F. P. Hudson, Concordance, isotopy, and diffeotopy, Ann. of Math. (2) 91 (1970), 425 – 448. · Zbl 0202.54602 · doi:10.2307/1970632
[4] L. S. Husch and I. Ivanšić, On shape concordance, Shape Theory and Geometric Topology , Lecture Notes in Math., vol. 870, Springer-Verlag, New York, 1981, pp. 135-149.
[5] I. Ivanšić and R. B. Sher, A complement theorem for continua in a manifold, The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979), 1979, pp. 437 – 452 (1980). · Zbl 0458.57009
[6] R. B. Sher, Complement theorems in shape theory, Shape theory and geometric topology (Dubrovnik, 1981) Lecture Notes in Math., vol. 870, Springer, Berlin-New York, 1981, pp. 150 – 168. · Zbl 0494.57007
[7] John Stallings, The embedding of homotopy types into manifolds, mimeographed notes. · Zbl 1246.57049
[8] Gerard A. Venema, Embeddings in shape theory, Shape theory and geometric topology (Dubrovnik, 1981) Lecture Notes in Math., vol. 870, Springer, Berlin-New York, 1981, pp. 169 – 185. · Zbl 0491.57009
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