MATH527_HW11 - Serge Ballif MATH 527 Homework 11 Problem 1...

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Serge Ballif MATH 527 Homework 11 November 28, 2007 Problem 1. Calculate the homology groups of S 3 with two unlinked circles deleted. Call our space X and the two deleted circles A and B . We can view a space homoeomorphic to the one above, by selecting any point of A on S 3 and stere- ographically projecting from that point into 3 . The stereographic image of A is then a line in 3 , and the stereographic image of B is an ellipsoid (homeo- morphic to a circle) that does not go around the image of A . We know that 3 with a line deleted deformation retracts onto a cylinder. 3 with a circle deleted deformation retracts onto the wedge of a sphere and a circle (from a pre- vious homework problem). Our space deformation retracts onto a combination of these, namely a wedge of a cylinder, sphere, and circle. The space further deformation retracts to the wedge of two circles and a sphere (since the cylinder deformation retracts to a circle). Hence, our space X will have the same homology groups as S 1 S 1 S 2 .
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  • Fall '07
  • ROTMAN,REGINA
  • Math, Topology, Topological space, homology groups, deformation retracts

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