# Munkres Acyclic Carrier

#### Socrates' Pen

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)
[itex]
\phi
[\latex]

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#### Landau

You have to end the latex-tag with /tex instead of \tex.

#### Socrates' Pen

THanks! Haha I was getting slightly annoyed, a simple mistake.
Now people can read it to help =)

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