{"id":110,"date":"2013-02-23T02:51:48","date_gmt":"2013-02-23T02:51:48","guid":{"rendered":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/?p=110"},"modified":"2013-07-16T13:33:15","modified_gmt":"2013-07-16T13:33:15","slug":"homework-7","status":"publish","type":"post","link":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/2013\/02\/23\/homework-7\/","title":{"rendered":"Homework 7"},"content":{"rendered":"<p>This is a two week assignment, due March 7.  This assignment is <strong>finally complete<\/strong>.<\/p>\n<p>A. Consider the sequence of homomorphisms \\(0 \\to \\mathbb{Z}^2 \\xrightarrow{g} \\mathbb{Z}^4 \\xrightarrow{f} \\mathbb{Z}^3 \\to 0 \\), where in the standard bases of \\(\\mathbb{Z}^n\\), \\(g = \\begin{pmatrix} 2 &#038; 0 \\\\ 0 &#038; 2 \\\\ -4 &#038; 8 \\\\ 2 &#038; -4 \\end{pmatrix}\\) and \\(f = \\begin{pmatrix} -3 &#038; 6 &#038; -1 &#038; 1\\\\ 1 &#038; -2 &#038; 2 &#038; 3 \\\\ 2 &#038; -4 &#038; 5 &#038; 8 \\end{pmatrix} \\).  Verify that this is a chain complex, and compute the homology groups at each level.  Specify generators for the homology groups in terms of the standard bases for the \\(\\mathbb{Z}^n\\)&#8217;s in terms of which \\(g\\) and \\(f\\) are given.<\/p>\n<p>B. Let \\(X\\) be a point.  Find the simplicial homology groups \\(H_n^\\Delta(X)\\), for \\(n\\geq 0\\).<\/p>\n<p>C. Let \\(X\\) be the Klein bottle.  Use the \\(\\Delta\\)-complex structure on page 102 of Hatcher to compute \\(H_n^\\Delta(X)\\), for \\(n\\geq 0\\).  Identify the generators of the homology groups with linear combinations of simplices.<\/p>\n<p>D. Recall that for a covering space \\(p: E \\to B\\), the map induced on the fundamental groups is a monomorphism.  In light of Hurewicz&#8217; Theorem, there is a map induced on first homology.  Does the map on homology need to be a monomorphism?<\/p>\n<p>E. Hatcher Section 2.1 number 8 (page 131).  You will get 3\/5 points for picking \\(n=5\\), and full points for the general case.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a two week assignment, due March 7. This assignment is finally complete. A. Consider the sequence of homomorphisms \\(0 \\to \\mathbb{Z}^2 \\xrightarrow{g} \\mathbb{Z}^4 \\xrightarrow{f} \\mathbb{Z}^3 \\to 0 \\), where in the standard bases of \\(\\mathbb{Z}^n\\), \\(g = \\begin{pmatrix} 2 &#038; 0 \\\\ 0 &#038; 2 \\\\ -4 &#038; 8 \\\\ 2 &#038; -4 [&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\/110"}],"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=110"}],"version-history":[{"count":18,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/110\/revisions"}],"predecessor-version":[{"id":201,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/110\/revisions\/201"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/media?parent=110"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/categories?post=110"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/tags?post=110"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}