Topology Preliminary Exam Solutions

The purpose of this part of the blog is to collect solutions to the past Topology preliminary examinations used at UAlbany (see this page). A page will be devoted to each test, although the existence of such a page does not necessarily imply that each problem on each test will be solved (e.g.,  given my present knowledge of Algebraic Topology, I don’t feel comfortable enough publishing solutions). Needless to say, if you find a problem on one of the exams that I haven’t solved, please do not post a solution in the comments, as I would like to discover it myself. However, please feel more than welcome to comment on problems for which I provided a solution, especially if you happen to find an error in my solution or know of a more succinct/elegant solution.