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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Deleted products of spaces which are unions of two simplexes


Author: W. T. Whitley
Journal: Trans. Amer. Math. Soc. 157 (1971), 99-111
MSC: Primary 57C05; Secondary 55D15
MathSciNet review: 0358792
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Abstract: If $ X$ is a space, the deleted product space, $ {X^ \ast }$, is $ X \times X - D$, where $ D$ is the diagonal. If $ Y$ is a space and $ f$ is a continuous map from $ X$ to $ Y$, then $ X_f^ \ast $ is the inverse image of $ {Y^ \ast }$ under the map $ f \times f$ taking $ X \times X$ into $ Y \times Y$. In this paper, we investigate the following questions: ``What maps $ f$ are such that $ X_f^ \ast $ is homotopically equivalent to $ {X^ \ast }$", and ``What maps $ f$ are such that $ X_f^ \ast $ is homotopically equivalent to $ f{(X)^ \ast }$?'' If $ X$ is the union of two nondisjoint simplexes and $ f$ is a simplicial map from $ X \times X$ such that $ f\vert f(X)$ is one-to-one, we obtain necessary and sufficient conditions for $ X_f^ \ast $ and $ f{(X)^ \ast }$ to be homotopically equivalent. If $ X$ is the union of nondisjoint simplexes $ A$ and $ B$ with $ \dim B = 1 + \dim (A \cap B)$, we obtain necessary and sufficient conditions for $ {X^ \ast }$ and $ X_f^ \ast $ to be homotopically equivalent if $ f$ is in the class of maps mentioned.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0358792-2
PII: S 0002-9947(1971)0358792-2
Keywords: Deleted products of spaces, unions of two simplexes, homotopy types of deleted products, homology groups of deleted products, simplicial maps on polyhedra, subspace of deleted products determined by simplicial map
Article copyright: © Copyright 1971 American Mathematical Society