Read Examples 2.36 to 2.43 in the book. We did not have time to cover all of them in class.<\/p>\n
Turn in 3 problems from the ones below<\/p>\n
Disclaimer: You should do all of the suggested problems on your own, and I will assume that you do. I would recommend (for the midterm, final, and qualifying exams) that you do many other problems. I am available for help.<\/p>\n
Due Wednesday, April 15.<\/p>\n
p 155 # 7, 9abd, 10, 14, 23, 28, 29, 31, <\/p>\n
B. Let \\(X = S^1 \\vee S^1\\), the circles labeled \\(a\\) and \\(b\\), and form \\(Y\\) by gluing two 2-cells onto \\(X\\) by the identifications \\(a^4\\) and \\(a^4b^{-2}a^2b^2a^{-2}\\). Compute the homology groups of \\(Y\\).<\/p>\n
C. Recall that \\(\\mathbb{R}P^3 = e^0\\cup e^1 \\cup e^2 \\cup e^3\\). Let \\(X\\) be the space obtained by attaching a 4-cell to \\(\\mathbb{R}P^3\\) where the composition of the quotient map and attaching map \\(\\Delta: S^3 \\to \\mathbb{R}P^3 \\to \\overline{e^3}\/\\partial{e^3} \\cong S^3\\) has degree \\(3\\). Compute the homology groups of \\(X\\).<\/p>\n
D. Recall that \\(\\mathbb{R}P^3\\) is naturally a subcomplex of \\(\\mathbb{R}P^4\\). Compute the homology of \\(\\mathbb{R}P^4\/ \\mathbb{R}P^3\\) using cellular homology.<\/p>\n
E. 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 – K\\) the knot complement. Compute \\(H_n(S^3 – K)\\) for all \\(n\\) using Mayer-Vietoris. Carefully specify generators of all spaces involved.<\/p>\n","protected":false},"excerpt":{"rendered":"
Read Examples 2.36 to 2.43 in the book. We did not have time to cover all of them in class. Turn in 3 problems from the ones below Disclaimer: You should do all of the suggested problems on your own, and I will assume that you do. I would recommend (for the midterm, final, and […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[2],"tags":[],"_links":{"self":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts\/91"}],"collection":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/comments?post=91"}],"version-history":[{"count":6,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts\/91\/revisions"}],"predecessor-version":[{"id":100,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts\/91\/revisions\/100"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/media?parent=91"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/categories?post=91"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/tags?post=91"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}