Embeddings of locally connected compacta

Venema, Gerard A.

doi:10.1090/S0002-9947-1985-0808741-5

Embeddings of locally connected compacta
HTML articles powered by AMS MathViewer

by Gerard A. Venema

Trans. Amer. Math. Soc. 292 (1985), 613-627

DOI: https://doi.org/10.1090/S0002-9947-1985-0808741-5

PDF | Request permission

Abstract:

Let $X$ be a $k$-dimensional compactum and $f:X \to {M^n}$ a map into a piecewise linear $n$-manifold, $n \geqslant k + 3$. The main result of this paper asserts that if $X$ is locally $(2k - n)$-connected and $f$ is $(2k - n + 1)$-connected, then $f$ is homotopic to a CE equivalence. In particular, every $k$-dimensional, $r$-connected, locally $r$-connected compactum is CE equivalent to a compact subset of ${{\mathbf {R}}^{2k - r}}$ as long as $r \leqslant k - 3$.

References

Morton Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478–483. MR 115157, DOI 10.1090/S0002-9939-1960-0115157-4
T. A. Chapman, Cell-like mappings, Algebraic and geometrical methods in topology (Conf. Topological Methods in Algebraic Topology, State Univ. New York, Binghamton, N.Y., 1973), Lecture Notes in Math., Vol. 428, Springer, Berlin, 1974, pp. 230–240. MR 0383423
Donald Coram and Ivan Ivanšić, Shape embedding theorem in (co) dimension two, Bull. Polish Acad. Sci. Math. 34 (1986), no. 1-2, 107–110 (English, with Russian summary). MR 850321
P. F. Duvall and L. S. Husch, A continuum of dimension $n$ which does not embed up to shape in $2n$-space, Proceedings of the International Conference on Geometric Topology (Warsaw, 1978) PWN, Warsaw, 1980, pp. 113–119. MR 656724
Jerzy Dydak and Jack Segal, Shape theory, Lecture Notes in Mathematics, vol. 688, Springer, Berlin, 1978. An introduction. MR 520227
Robert D. Edwards, Demension theory. I, Geometric topology (Proc. Conf., Park City, Utah, 1974) Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975, pp. 195–211. MR 0394678
Steve Ferry, A stable converse to the Vietoris-Smale theorem with applications to shape theory, Trans. Amer. Math. Soc. 261 (1980), no. 2, 369–386. MR 580894, DOI 10.1090/S0002-9947-1980-0580894-1
Steve Ferry, Homotopy, simple homotopy and compacta, Topology 19 (1980), no. 2, 101–110. MR 572578, DOI 10.1016/0040-9383(80)90001-4
J. F. P. Hudson, Piecewise linear embeddings, Ann. of Math. (2) 85 (1967), 1–31. MR 215308, DOI 10.2307/1970522
L. S. Husch, $UV^{k}$ maps and embeddings, Topology Appl. 15 (1983), no. 3, 273–291. MR 694547, DOI 10.1016/0166-8641(83)90058-5
L. S. Husch and I. Ivanšić, Embedding compacta up to shape, Shape theory and geometric topology (Dubrovnik, 1981) Lecture Notes in Math., vol. 870, Springer, Berlin-New York, 1981, pp. 119–134. MR 643527
M. C. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965), 1–14. MR 182978, DOI 10.2307/1970560
R. C. Lacher, Cell-like mappings. I, Pacific J. Math. 30 (1969), 717–731. MR 251714
Vo Thanh Liem and Gerard A. Venema, Stable shape concordance implies homeomorphic complements, Fund. Math. 126 (1986), no. 2, 123–131. MR 843241, DOI 10.4064/fm-126-2-123-131

The embedding of homotopy types into manifolds

M. A. Štan′ko, Imbedding of compacta in euclidean space, Mat. Sb. (N.S.) 83 (125) (1970), 234–255 (Russian). MR 0271923
Hassler Whitney, The self-intersections of a smooth $n$-manifold in $2n$-space, Ann. of Math. (2) 45 (1944), 220–246. MR 10274, DOI 10.2307/1969265