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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Embeddings of locally connected compacta


Author: Gerard A. Venema
Journal: Trans. Amer. Math. Soc. 292 (1985), 613-627
MSC: Primary 57Q35; Secondary 54C10, 54F35, 57N15, 57N25, 57N60
DOI: https://doi.org/10.1090/S0002-9947-1985-0808741-5
MathSciNet review: 808741
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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 $r \leqslant k - 3$.


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Keywords: Cell-like set, compactum, CE map, CE equivalence, topological embedding, <IMG WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$r$">-connected, locally <IMG WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$r$">-connected
Article copyright: © Copyright 1985 American Mathematical Society