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kakarotyjn

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I'm reading Allen Hatcher's topology book.In order to prove a theorem about homotopic maps induce the same homomorphism of homology groups,given a homotopy [tex]F:X \times I \to Y[/tex] from f to g,the author construct a prism operators

[tex]P:C_n (X) \to C_{n + 1} (Y)[/tex] by [tex]P(\sigma ) = \sum\nolimits_i {( - 1)^i F \circ (\sigma \times 1)|[v_0 ,...,v_i ,w_i ,...,w_n ]} [/tex] for [tex]\sigma :\Delta ^n \to X[/tex],where [tex]{F \circ (\sigma \times 1)}[/tex] is the composition [tex]\Delta ^n \times I \to X \times I \to Y[/tex].

I don't understand how sigma*1 acts on the n+1 simplex,sigma acts on n simplex,what the 1 acts on?Why[tex]F \circ (\sigma \times 1)|[\mathop v\limits^ \wedge _0 ,w_0 ,...,w_n ][/tex] equals to [tex]g \circ \sigma = g_\# (\sigma )[/tex]

Need helps,thank you!

[tex]P:C_n (X) \to C_{n + 1} (Y)[/tex] by [tex]P(\sigma ) = \sum\nolimits_i {( - 1)^i F \circ (\sigma \times 1)|[v_0 ,...,v_i ,w_i ,...,w_n ]} [/tex] for [tex]\sigma :\Delta ^n \to X[/tex],where [tex]{F \circ (\sigma \times 1)}[/tex] is the composition [tex]\Delta ^n \times I \to X \times I \to Y[/tex].

I don't understand how sigma*1 acts on the n+1 simplex,sigma acts on n simplex,what the 1 acts on?Why[tex]F \circ (\sigma \times 1)|[\mathop v\limits^ \wedge _0 ,w_0 ,...,w_n ][/tex] equals to [tex]g \circ \sigma = g_\# (\sigma )[/tex]

Need helps,thank you!

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