By Monday, 26 January, please read all of Section 1.1 in Hatcher, especially the section on Induced Homomorphisms p. 34-37. (I did not finish my lecture on the 21st. What I didn’t cover was in Induced Homomorphisms).<\/p>\n
Due Wednesday, 28 January, in class.<\/p>\n
Select and turn in three problems from Section 1.1, p. 38-40 and from the supplementary problems A, B, and C, below.<\/p>\n
Suggested problems: #1, 5, 8, 11, 14, 16, 17, A, B, C<\/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
A. Let \\( f: (X,x) \\to (Y,y) \\) be a continuous map. \u00a0Show that the induced map \\( f_\\ast: \\pi_1(X, x) \\to \\pi_1(Y,y) \\) is a group homomorphism. \u00a0(I showed you the steps in class, please provide justification).<\/p>\n
B. Let \\( (X,x) \\overset{f}{\\to} (Y,y) \\overset{g}{\\to} (Z,z) \\) be maps with \\(y=f(x)\\) and \\(z=g(y)\\). \u00a0Show that the induced maps \\( \\pi_1(X,x) \\overset{f_\\ast}{\\to} \\pi_1(Y,y) \\overset{g_\\ast}{\\to} \\pi_1(Z,z) \\)\u00a0satisfy \\(g_\\ast \\circ f_\\ast = (g\\circ f)_\\ast \\)<\/p>\n
C. For any pointed space \\((X,x)\\), show that \\((\\mathrm{id}_X)_\\ast = \\mathrm{id}_{\\pi_1(X,x)} \\)<\/p>\n
(From B and C you can deduce that a homeomorphism induces an isomorphism on the fundamental group)<\/p>\n","protected":false},"excerpt":{"rendered":"
By Monday, 26 January, please read all of Section 1.1 in Hatcher, especially the section on Induced Homomorphisms p. 34-37. (I did not finish my lecture on the 21st. What I didn’t cover was in Induced Homomorphisms). Due Wednesday, 28 January, in class. Select and turn in three problems from Section 1.1, p. 38-40 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\/28"}],"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=28"}],"version-history":[{"count":8,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts\/28\/revisions"}],"predecessor-version":[{"id":36,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts\/28\/revisions\/36"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/media?parent=28"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/categories?post=28"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/tags?post=28"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}