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Math Help - a homotopy between two finite sets of distinct cardinality

  1. #1
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    a homotopy between two finite sets of distinct cardinality

    Dear all;

    Let us discuss this problem:

    Let A and B be two great circles in a sphere S^2={x^2+y^2+z^2=1} such that A intersects B in two and only two points (recall that any two great circles intersect exactly twise). Let X be a finite set of points in A such that the arc lenght between each two of these points is equal and the cardinality of X is prime, i.e X contains prime number of elements.. For example if X contains 5 points, then the arc between each two of the 5 points subtending an angle 72 degree. Similarly, let Y be a finite set of points in B with equal arc length between each two points. Suppose also #B is prime and distinct from #A.

    The question is: Can we define a homotopy between X and Y so that X is homotopic to Y. If so, how

    Thank you in advance
    Last edited by student2011; March 11th 2014 at 10:43 PM.
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  2. #2
    Junior Member
    Joined
    Feb 2011
    Posts
    72

    Re: a homotopy between two finite sets of distinct cardinality

    Sorry, my question above is not clear, here I rewrite it to be more accurate and specific:

    Let A={a_{1},a_{2},...,a_{n}}, B={b_{1},b_{2},...,b_{n}} and C={c_{1},c_{2},..,c_{n-1}} be discrete finite sets embedded in a unit sphere S^2=x^2+y^2+z^2=1 so that the topologies of A,B and C are induced topology from the topology of the sphere (Note that the topologies of A, B and C are P(A), P(B) and P(C) respectively, where P(A) denotes the power set of A). Assume also that the points of B are imbedded in a great circle in S^2 such that the arc length between each two of these points is equal. Also suppose that elements of C are imbedded in a great circle (It can be same great circle of B) so that the arc length between each two of these elements is equal. Define f_{0} from A to B by f_{0}(a_{i})=b_{i} for i=1,2,...,n. Define also a function f_{1} from A to C by f_{1}(a_{1})=f_{1}(a_{n})=c_{1} and f_{1}(a_{i})=c_{i} otherwise. The question is: Is it possible to define a homotopy F from A * [0,1] to S^{2} between f_{0} and f_{1}.

    Thank you in advance
    Last edited by student2011; March 17th 2014 at 06:42 AM.
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