Qualifying Exam in Topology

Fall 1998

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1. Prove, by a carefully worked-out example, that if and are topological spaces, the homology group is not necessarily isomorphic to the direct sum of and .
   
   
2. Let , , be Hausdorf, path-connected, locally path-connected. Let
   

   be maps, with .

   1. Prove that if and are covering space projections then is too.
   2. Prove that if has a universal covering space, and if , are covering space projections, then is too.
   
   
   
3. Construct, for each and every a map of degree .
   
   
4. (a) State the Eilenberg-Steenrod axioms for a homology theory. (b) State the Universal Coefficient Theorem, and give a non-trivial example.
   
   
5. Prove that every differentiable -manifold can be embedded in Euclidean space of some dimension .