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This is theorem 13.6 in Munkres' Elements of Algebraic Topology. I'm trying to go through this, but I can't prove it. Can someone do this one please?

Btw, its "Choose a partial ordering of the vertices of [tex]K[/tex] that induces a linear ordering on the vertices of each simplex of [tex]K[/tex]. Define [tex]\phi:C_p(K)\to C_p'(K)[/tex] by letting [tex]\[\phi([v_0,...,v_p])=(v_0,...,v_p)\][/tex] if [tex]v_0<v_1<...<v_p[/tex] in the given ordering. Define [tex]\psi:C_p'(K)\to C_p(K)[/tex] by

[tex] \[\psi((w_0,...,w_p))=\begin{cases}[w_0,...,w_p] & \text{if the }w_i \text{ are distinct} \\ 0 * \text{otherwise} \end{cases} \] [/tex]

Then [tex]\phi,\psi[/tex] are augmentation-preserving chain maps that are chian homotopy inverses.

Thanks!!! The sooner the better of course, much appreciated. (Apply the acyclic carrier theorem)

[itex]

\phi

[\latex]

Btw, its "Choose a partial ordering of the vertices of [tex]K[/tex] that induces a linear ordering on the vertices of each simplex of [tex]K[/tex]. Define [tex]\phi:C_p(K)\to C_p'(K)[/tex] by letting [tex]\[\phi([v_0,...,v_p])=(v_0,...,v_p)\][/tex] if [tex]v_0<v_1<...<v_p[/tex] in the given ordering. Define [tex]\psi:C_p'(K)\to C_p(K)[/tex] by

[tex] \[\psi((w_0,...,w_p))=\begin{cases}[w_0,...,w_p] & \text{if the }w_i \text{ are distinct} \\ 0 * \text{otherwise} \end{cases} \] [/tex]

Then [tex]\phi,\psi[/tex] are augmentation-preserving chain maps that are chian homotopy inverses.

Thanks!!! The sooner the better of course, much appreciated. (Apply the acyclic carrier theorem)

[itex]

\phi

[\latex]

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