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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$.

References [Enhancements On Off] (What's this?)

  • [1] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Springer-Verlag, Berlin and New York, 1964. MR 0176478 (31:750)
  • [2] C. W. Patty, A note on the homology of deleted product spaces, Proc. Amer. Math. Soc. 14 (1963), 800. MR 0155322 (27:5256)
  • [3] C. T. Yang, On theorems of Borsuk-Ulam Kakutani-Yamabe-Yujobô and Dyson. I, Ann. of Math. (2) 60 (1954), 262-282. MR 0065910 (16:502d)

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Article copyright: © Copyright 1995 American Mathematical Society

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