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, February 25.<\/p>\n
Suggested problems:<\/p>\n
Hatcher 1.3.17 and 18<\/p>\n
A. On p 58, pick one of the covers from (7) – (14) and find \\(\\mathrm{Aut}(p)\\). Is it a normal cover?<\/p>\n
B. Let \\(p:\\widetilde{X}\\to X\\) be a covering with \\( \\widetilde{x}, \\widetilde{x}’ \\in p^{-1}(x)\\). Let \\(G = \\pi_1(X,x)\\) and \\(H = p_\\ast(\\pi_1(\\widetilde{X},\\widetilde{x}))\\). Let \\(N(H) \\) denote the normalizer of \\(H\\), i.e. the largest subgroup of \\(G\\) in which \\(H\\) is normal. Let \\(\\widetilde{\\gamma}\\) be a path in \\(\\widetilde{X}\\) from \\(\\widetilde{x}\\) to \\(\\widetilde{x}’\\). Prove if \\(p_\\ast([\\widetilde{\\gamma}]) \\in N(H)\\), then there exists an \\(f\\in \\mathrm{Aut}(p)\\) such that \\(f(\\widetilde{x}) = \\widetilde{x}’\\). Is the converse true?<\/p>\n
C. Let \\(f:\\mathbb{Z}^4 \\to \\mathbb{Z}^3\\) be a linear transformation. Let \\(\\{a,b,c,d\\}\\) and \\(\\{x,y,z\\}\\) be bases for the domain and codomain respectively. Suppose that the matrix representing \\(f\\) with respect to these basis is \\( \\begin{pmatrix} 7 & -14 & 24 & 41\\\\ 2 & -4 & 4 & 6 \\\\ 4 & -8 & 10 & 16 \\end{pmatrix} \\)<\/p>\n
Find the groups \\( \\mathrm{im}(f), \\mathrm{ker}(f), \\mathbb{Z}^4\/\\mathrm{ker}(f), \\mathrm{\\ and\\ } \\mathrm{coker}(f) \\) and give generators for them in terms of \\(a,b,c,d,x,y,z\\).<\/p>\n
D. Any problem from Homework 4 or 5 that you or I didn’t already do.<\/p>\n","protected":false},"excerpt":{"rendered":"
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 qualifying exams) that you do many other problems. I am available for help. Due Wednesday, February 25. Suggested problems: Hatcher […]<\/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\/57"}],"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=57"}],"version-history":[{"count":10,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts\/57\/revisions"}],"predecessor-version":[{"id":71,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts\/57\/revisions\/71"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/media?parent=57"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/categories?post=57"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/tags?post=57"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}