This will be due March 21.<\/p>\n
A. Let \\(A\\) and \\(B\\) be abelian groups and \\(f\\) a homomorphism. Show that \\(0\\to A\\xrightarrow{f} B\\) is exact if and only if \\(f\\) is injective.<\/p>\n
B. Suppose \\(\\cdots \\to A_{n+1} \\xrightarrow{f_{n+1}} A_n \\xrightarrow{f_n} A_{n-1} \\to \\cdots\\) is a long exact sequence. Prove that for each \\(n\\) there exists a short exact sequence \\(0\\to\\mathrm{coker\\ } f_{n+2} \\xrightarrow{\\phi} A_n \\xrightarrow{\\theta} \\mathrm{ker\\ }f_{n-1} \\to 0\\), where \\(\\phi\\) is induced from \\(f_{n+1}\\) and \\(\\theta\\) is induced from \\(f_n\\).<\/p>\n
C. Suppose \\(A \\subset X\\), with \\(i:A\\to X\\) the inclusion map, and suppose there is a retraction \\(r:X\\to A\\). Prove that the induced map \\(i_\\ast: H_n(A)\\to H_n(X)\\) is a monomorphism and that the induced map \\(r_\\ast: H_n(X)\\to H_n(A)\\) is an epimorpishm, for all \\(n\\).<\/p>\n
D. Let \\(X\\) be a topological space and suppose \\(f:X\\to X\\) is a constant map. Show that the induced map on reduced homology \\(f_\\ast:\\widetilde{H}_n(X)\\to \\widetilde{H}_n(X) \\) is the zero map for all \\(n\\).<\/p>\n
E. Suppose \\(A\\) is an abelian group and that \\(0 \\to \\mathbb{Z} \\xrightarrow{\\times 5} \\mathbb{Z} \\to A \\to \\mathbb{Z} \\to \\mathbb{Z} \\to 0\\) is exact. Classify \\(A\\).<\/p>\n
F. Let \\(x_0\\) be a point in a space \\(X\\). Show that \\(H_0(X,x_0) \\cong \\widetilde{H}_0(X)\\). Hint: play with the quotient \\( Z_0(X,x_0) \/ B_0(X,x_0) \\), and recall the theorem involving \\(\\widetilde{H}_0(X)\\).<\/p>\n
Problems G and H relate to the Snake Lemma. Suppose we have the following commutative diagram of abelian groups where the rows are exact:<\/p>\n
\\( \\newcommand{\\ra}[1]{\\kern-1.5ex\\xrightarrow{\\ \\ #1\\ \\ }\\phantom{}\\kern-1.5ex} \\newcommand{\\ras}[1]{\\kern-1.5ex\\xrightarrow{\\ \\ \\smash{#1}\\ \\ }\\phantom{}\\kern-1.5ex} \\newcommand{\\da}[1]{\\bigg\\downarrow\\raise.5ex\\rlap{\\scriptstyle#1}} \\begin{array}{c} & & M’ & \\ra{f} & M & \\ra{g} & M” & \\ra{ } & 0 \\\\ & & \\da{d’} & & \\da{d} & & \\da{d”} & & \\\\ 0 & \\ras{ } & N’ & \\ras{f’} & N & \\ras{g’} & N” & & \\\\ \\end{array} \\)
\nThe Snake Lemma says that the sequence \\( \\mathrm{ker\\ } d’ \\xrightarrow{\\overline{f}} \\mathrm{ker\\ } d \\xrightarrow{\\overline{g}} \\mathrm{ker\\ } d”\\xrightarrow{\\partial} \\mathrm{coker\\ } d’ \\xrightarrow{\\overline{f’}} \\mathrm{coker\\ } d \\xrightarrow{\\overline{g’}} \\mathrm{coker\\ } d” \\) is exact, where \\(\\partial = (f’)^{-1}\\circ d \\circ g^{-1}\\).<\/p>\n
G. Prove the connecting map \\(\\partial\\) is well-defined.<\/p>\n
H. Show \\(\\mathrm{im\\ }\\partial = \\mathrm{ker\\ } \\overline{f’}\\).<\/p>\n","protected":false},"excerpt":{"rendered":"
This will be due March 21. A. Let \\(A\\) and \\(B\\) be abelian groups and \\(f\\) a homomorphism. Show that \\(0\\to A\\xrightarrow{f} B\\) is exact if and only if \\(f\\) is injective. B. Suppose \\(\\cdots \\to A_{n+1} \\xrightarrow{f_{n+1}} A_n \\xrightarrow{f_n} A_{n-1} \\to \\cdots\\) is a long exact sequence. Prove that for each \\(n\\) there exists […]<\/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\/131"}],"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=131"}],"version-history":[{"count":27,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/131\/revisions"}],"predecessor-version":[{"id":200,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/posts\/131\/revisions\/200"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/media?parent=131"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/categories?post=131"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/spring2013mat761\/wp-json\/wp\/v2\/tags?post=131"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}