This is due March 28.<\/p>\n
0. Read section starting p 128 on equivalence of singular and simplicial homology. Now you know it and may use it.<\/p>\n
A. Let \\(A\\) be a subspace of \\(X\\). Show that \\(H_0(X,A) = 0\\) iff \\(A\\) meets each path-component of \\(X\\).<\/p>\n
B. Let \\(A\\) be a subspace of \\(X\\). Show that \\(H_1(X,A) = 0\\) iff \\(H_1(A)\\to H_1(X)\\) is surjective and each path-component of \\(X\\) contains at most one path-component of \\(A\\).<\/p>\n
C. Let \\(A\\) be a subspace of \\(X\\). Show that \\(i_\\ast: H_n(A) \\to H_n(X)\\) is an isomorphism for all \\(n\\) iff \\(H_n(X,A) = 0 \\) for all \\(n\\).<\/p>\n
D. Find \\(\\widetilde{H}_n( S^1\\times S^1 \/ \\ast \\times S^1 ) \\) for all \\(n\\) using the long exact sequence for the homology of a good pair. Here you might need to use the generators of simplicial homology, but you need not specify generators for the homology groups of the quotient.<\/p>\n
E. Suppose we have a smooth embedding \\(f: S^1\\times D^2 \\hookrightarrow \\mathbb{R}^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 \\(\\mathbb{R}^3 – K\\) the knot complement. Compute \\(H_n(\\mathbb{R}^3 – K)\\) for all \\(n\\). Hint: use homotopy invariance and excision, and argue that \\(H_n(\\mathbb{R}^3, \\mathbb{R}^3 – N(K)) \\cong H_n(\\mathbb{R}^3, \\mathbb{R}^3-K)\\).<\/p>\n
F. Find \\(H_n(S^1 \\vee (S^1\\times S^1) \\vee S^2 \\vee S^4)\\) for all \\(n\\) and specify generators. Draw a picture.<\/p>\n","protected":false},"excerpt":{"rendered":"
This is due March 28. 0. Read section starting p 128 on equivalence of singular and simplicial homology. Now you know it and may use it. A. Let \\(A\\) be a subspace of \\(X\\). Show that \\(H_0(X,A) = 0\\) iff \\(A\\) meets each path-component of \\(X\\). B. Let \\(A\\) be a subspace of \\(X\\). Show […]<\/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\/160"}],"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=160"}],"version-history":[{"count":7,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/160\/revisions"}],"predecessor-version":[{"id":199,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/160\/revisions\/199"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/media?parent=160"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/categories?post=160"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/tags?post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}