This homework assignment is due Thursday, January 31, in class.<\/p>\n
Hatcher Section 1.1, p.38 # 1, 5, 8, 11, 16. Section 1.2 #\u00a03, 4, 8, 19.<\/p>\n
Then<\/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(After some comments made in class, it suffices to prove that for loops \\(\\phi\\) and \\(\\psi\\), we have the equality of maps\/loops \\( f(\\phi\\cdot\\psi) = f(\\phi)\\cdot f(\\psi) \\)<\/p>\n
B. Let \\( (X,x) \\xrightarrow{f} (Y,y) \\xrightarrow{g} (Z,z) \\) be maps. \u00a0Show that the induced maps \\( \\pi_1(X,x) \\xrightarrow{f_\\ast} \\pi_1(Y,y) \\xrightarrow{g_\\ast} \\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
D. Classify the letters A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z up to homotopy type. \u00a0Little justification is needed.<\/p>\n","protected":false},"excerpt":{"rendered":"
This homework assignment is due Thursday, January 31, in class. Hatcher Section 1.1, p.38 # 1, 5, 8, 11, 16. Section 1.2 #\u00a03, 4, 8, 19. Then 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. […]<\/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\/67"}],"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=67"}],"version-history":[{"count":11,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/67\/revisions"}],"predecessor-version":[{"id":206,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/67\/revisions\/206"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/media?parent=67"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/categories?post=67"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/tags?post=67"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}