Mathematics – Geometric Topology
Scientific paper
2010-03-25
Topology and its Applications 157, 10-11 (2010) 1742-1759
Mathematics
Geometric Topology
31 pages
Scientific paper
10.1016/j.topol.2010.02.024
Let (X, t, S) be a triple, where S is a compact, connected surface without boundary, and t is a free cellular involution on a CW-complex X. The triple (X, t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map f:X-->S, there exists a point x belonging to X satisfying f(t(x))=f(x). In this paper, we formulate this property in terms of a relation in the 2-string braid group B_2(S) of S. If X is a compact, connected surface without boundary, we use this criterion to classify all triples (X, t, S) for which the Borsuk-Ulam property holds. We also consider various cases where X is not necessarily a surface without boundary, but has the property that \pi_1(X/t) is isomorphic to the fundamental group of such a surface. If S is different from the 2-sphere S^2 and the real projective plane RP^2, then we show that the Borsuk-Ulam property does not hold for (X, t, S) unless either \pi_1(X/t) is isomorphic to \pi_1(RP^2), or \pi_1(X/t) is isomorphic to the fundamental group of a compact, connected non-orientable surface of genus 2 or 3 and S is orientable. In the latter case, the veracity of the Borsuk-Ulam property depends further on the choice of involution t; we give a necessary and sufficient condition for it to hold in terms of the surjective homomorphism \pi_1(X/t)-->Z_2 induced by the double covering X-->X/t. The cases S=S^2,RP^2 are treated separately.
Gonçalves Daciberg Lima
Guaschi John
No associations
LandOfFree
The Borsuk-Ulam theorem for maps into a surface does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Borsuk-Ulam theorem for maps into a surface, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Borsuk-Ulam theorem for maps into a surface will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-556480