{"id":181,"date":"2013-04-11T14:31:45","date_gmt":"2013-04-11T14:31:45","guid":{"rendered":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/?p=181"},"modified":"2013-07-16T13:33:15","modified_gmt":"2013-07-16T13:33:15","slug":"homework-12","status":"publish","type":"post","link":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/2013\/04\/11\/homework-12\/","title":{"rendered":"Homework 12"},"content":{"rendered":"<p>Due in class April 18.<\/p>\n<p>p 155, section 2.2: 21, 22, 28a, 29<\/p>\n<p>A. Suppose we have a smooth embedding \\(f: S^1\\times D^2 \\hookrightarrow S^3\\).  Let \\(K = f(S^1\\times 0)\\) and \\(N(K) = f(S^1\\times D^2)\\).  We call \\(K\\) a knot and \\(N(K)\\) its tubular neighborhood.  We call \\(S^3 &#8211; K\\) the knot complement.  Compute \\(H_n(S^3 &#8211; K)\\) for all \\(n\\) using Mayer-Vietoris.  Carefully specify generators of all spaces involved.<\/p>\n<p>B. Consider \\(\\mathbb{R}^3 &#8211; K \\subset S^3 &#8211; K \\).  Using a Mayer-Vietoris sequence, find a generator for \\(H_2(\\mathbb{R}^3-K)\\).<\/p>\n<p>C. Write a short essay outlining a proof that for a homology theory \\(h\\) on the category of finite CW pairs, the group \\(h_0(\\ast)\\) determines all homology groups \\(h_n(X,A)\\) for all CW pairs \\((X,A)\\).  A flow chart may be useful.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Due in class April 18. p 155, section 2.2: 21, 22, 28a, 29 A. Suppose we have a smooth embedding \\(f: S^1\\times D^2 \\hookrightarrow S^3\\). Let \\(K = f(S^1\\times 0)\\) and \\(N(K) = f(S^1\\times D^2)\\). We call \\(K\\) a knot and \\(N(K)\\) its tubular neighborhood. We call \\(S^3 &#8211; K\\) the knot complement. Compute \\(H_n(S^3 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[4],"tags":[],"_links":{"self":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/181"}],"collection":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/comments?post=181"}],"version-history":[{"count":10,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/181\/revisions"}],"predecessor-version":[{"id":196,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/181\/revisions\/196"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/media?parent=181"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/categories?post=181"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/tags?post=181"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}