Turn in 3 problems from Section 1.2 and the extra problems (A-D) below, and read Section 1.2 (it contains some great examples). Think about all the ways you know how to calculate \\(\\pi_1\\) of a space.<\/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 4.<\/p>\n
Suggested problems:<\/p>\n
Section 1.2 # 3, 4, 8, 10, 19<\/p>\n
A. The (initial) maps \\(\\varphi_\\alpha: G_\\alpha \\to G\\) when defining the free product \\(G\\) of the groups \\(\\{G_\\alpha\\}_{\\alpha\\in \\Lambda}\\) are injective.<\/p>\n
B. If the free product of groups exists, then it is unique up to isomorphism. (Use the universal property)<\/p>\n
C. \\(F(\\mathcal{A}) \\cong F(\\mathcal{B})\\) if and only if \\(\\#\\mathcal{A} = \\#\\mathcal{B}\\)<\/p>\n
D. The Klein bottle \\(K\\) can be given a CW structure consisting of one 0-cell, two 1-cells (labelled \\(a\\) and \\(b\\)), and one 2-cell attached via the loop \\(aba^{-1}b\\). I claimed in class that \\(K\\) is homeomorphic to some orientable or nonorientable surface in the list \\(\\ldots N_2, N_1, M_0, M_1, M_2 \\ldots\\) Which one is it? Prove it is homeomorphic to \\(K\\). Note: this will take some creativity, and I’m happy to give a hint.<\/p>\n","protected":false},"excerpt":{"rendered":"
Turn in 3 problems from Section 1.2 and the extra problems (A-D) below, and read Section 1.2 (it contains some great examples). Think about all the ways you know how to calculate \\(\\pi_1\\) of a space. Disclaimer: You should do all of the suggested problems on your own, and I will assume that you do. […]<\/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\/37"}],"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=37"}],"version-history":[{"count":6,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts\/37\/revisions"}],"predecessor-version":[{"id":44,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/posts\/37\/revisions\/44"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/media?parent=37"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/categories?post=37"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2015mat761\/wp-json\/wp\/v2\/tags?post=37"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}