What is the Acyclic Carrier Theorem and how can it be applied using chain maps?

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SUMMARY

The Acyclic Carrier Theorem, as discussed in Munkres' "Elements of Algebraic Topology," involves defining chain maps that preserve augmentation and serve as chain homotopy inverses. Specifically, the theorem states that by choosing a partial ordering of the vertices of a simplicial complex K, one can define the chain maps \(\phi\) and \(\psi\) that facilitate this relationship. The construction of these maps is crucial for proving the theorem, which is essential for understanding the topology of simplicial complexes.

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

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

Then \phi,\psi are augmentation-preserving chain maps that are chian homotopy inverses.
Thanks! The sooner the better of course, much appreciated. (Apply the acyclic carrier theorem)
<br /> \phi<br /> [\latex]
 
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You have to end the latex-tag with /tex instead of \tex.
 
THanks! Haha I was getting slightly annoyed, a simple mistake.
Now people can read it to help =)