This assignment is due April 11 in class.<\/p>\n
A. Let \\(M_2 \\) denote the orientable genus two surface, and let \\(X\\) denote \\(M_2 – \\mathrm{int\\ } D^2\\). Call the boundary curve \\(\\gamma\\). Form a space \\(Y\\) by attaching a disk \\(D^2\\) along its boundary to the curve \\(\\gamma\\) by a map of degree \\(5\\). Compute the homology groups of \\(Y\\).<\/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 \\xrightarrow{\\phi} \\mathbb{R}P^3 \\xrightarrow{q} \\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","protected":false},"excerpt":{"rendered":"
This assignment is due April 11 in class. A. Let \\(M_2 \\) denote the orientable genus two surface, and let \\(X\\) denote \\(M_2 – \\mathrm{int\\ } D^2\\). Call the boundary curve \\(\\gamma\\). Form a space \\(Y\\) by attaching a disk \\(D^2\\) along its boundary to the curve \\(\\gamma\\) by a map of degree \\(5\\). Compute […]<\/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\/173"}],"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=173"}],"version-history":[{"count":8,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/173\/revisions"}],"predecessor-version":[{"id":197,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/173\/revisions\/197"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/media?parent=173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/categories?post=173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/tags?post=173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}