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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Antipodal coincidence for maps of spheres into complexes

Authors: Marek Izydorek and Jan Jaworowski
Journal: Proc. Amer. Math. Soc. 123 (1995), 1947-1950
MSC: Primary 55M20; Secondary 55M35
MathSciNet review: 1242089
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Abstract: This paper gives a partial answer to the question of whether there exists a Borsuk-Ulam type theorem for maps of ${S^n}$ into lower-dimensional spaces, which are not necessarily manifolds. It is shown that for each k and $n \leq 2k - 1$, there exists a map f of ${S^n}$ into a contractible k-dimensional complex Y such that $fx \ne f( - x)$, for all $x \in {S^n}$. In particular, there exists a map of ${S^3}$ into a 2-dimensional complex Y without an antipodal coincidence. This answers a question raised by Conner and Floyd in 1964. The complex Y provides also an example of a countractible k-dimensional complex whose deleted product has the Yang-index equal to $2k - 1$.

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Article copyright: © Copyright 1995 American Mathematical Society