Antipodal coincidence for maps of spheres into complexes
Marek Izydorek and Jan Jaworowski
Proc. Amer. Math. Soc. 123 (1995), 1947-1950
Primary 55M20; Secondary 55M35
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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 into lower-dimensional spaces, which are not necessarily manifolds. It is shown that for each k and , there exists a map f of into a contractible k-dimensional complex Y such that , for all . In particular, there exists a map of 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 .
E. Conner and E.
E. Floyd, Differentiable periodic maps, Ergebnisse der
Mathematik und ihrer Grenzgebiete, N. F., Band 33, Academic Press Inc.,
Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg,
0176478 (31 #750)
W. Patty, A note on the homology of deleted
product spaces, Proc. Amer. Math. Soc. 14 (1963), 800. MR 0155322
(27 #5256), http://dx.doi.org/10.1090/S0002-9939-1963-0155322-6
Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and
Dyson. I, Ann. of Math. (2) 60 (1954), 262–282.
- P. E. Conner and E. E. Floyd, Differentiable periodic maps, Springer-Verlag, Berlin and New York, 1964. MR 0176478 (31:750)
- C. W. Patty, A note on the homology of deleted product spaces, Proc. Amer. Math. Soc. 14 (1963), 800. MR 0155322 (27:5256)
- 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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