MATH527_HW12 - Serge Ballif MATH 527 Homework 12 December 7...

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Serge Ballif MATH 527 Homework 12 December 7, 2007 Problem 1. Show that S n is not homeomorphic to any proper subspace of itself. Recall invariance of domain: Let U be open in n (or we could use S n ). Let f : U n be continuous and injective. Then f ( U ) is open in n and f is an imbedding. Let U be a subspace of S n that is homeomorphic to S n via a homeomorphism h (i.e. U = f ( S n )). Then U is compact as the continuous image of the compact space S n . Since S n is Hausdorff, U is closed. However, by invariance of domain, U is open. S n is connected, so the only subsets of S n that are both open and closed are S n and . Therefore U = S n . Therefore, there is no homeomorphism from S n to a proper subspace of itself. Problem 2. Let A be a closed subspace of n homeomorphic to k . Compute the homology groups of n - A . e H i ( n - A ) = for i = n - k - 1, and e H i ( n - A ) = 0 otherwise. To see this, we note that since A is closed, but not compact, it must be unbounded. Hence, we might as well assume that A = k . Then n - A deformation retracts to S n - 1 - S k - 1 , which is a space that has the desired homology groups.
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