Register to reply

Munkres Acyclic Carrier

by Socrates' Pen
Tags: acyclic, carrier, munkres
Share this thread:
Socrates' Pen
Mar13-11, 12:04 PM
P: 3
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)
Phys.Org News Partner Science news on
'Office life' of bacteria may be their weak spot
Lunar explorers will walk at higher speeds than thought
Philips introduces BlueTouch, PulseRelief control for pain relief
Mar13-11, 12:43 PM
Sci Advisor
P: 905
You have to end the latex-tag with /tex instead of \tex.
Socrates' Pen
Mar13-11, 12:48 PM
P: 3
THanks! Haha I was getting slightly annoyed, a simple mistake.
Now people can read it to help =)

Register to reply

Related Discussions
Pronunciation of Munkres General Math 9
Munkres' Topology Science & Math Textbooks 2
Munkres lemma 81.1 Calculus & Beyond Homework 1
Topology by munkres Science & Math Textbooks 3
What Is Acyclic Isomer Chemistry 1