My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
February 27th, 2009, 09:24 AM   #1
Member
 
Joined: Jan 2009

Posts: 72
Thanks: 0

Chain Homotopy Perplexity

Question:

Where does the minus sign come from in the definition of chain homotopy, for example as depicted in the Wikipedia entry "Homotopy category of chain complexes"?

I have trouble motivating the definition... It appears the diagram in the cited location is not intended to be commutative. Is that correct?

Any insights would be appreciated as I am trying to teach myself homology from a nice (concise) little book by Keesee. I also have Massey but his discussion isn't easy. (In fact, the book seems to fall short of Springer's usual quality in several ways, including the typesetting.)

I'm afraid I'm more of a "point set" guy. Any other sources recommended?

Thanks!

Note added in proof: As I write this I think I'm getting a clue: We say the maps f and g are chain homotopic IF the "difference" f-g can be expressed in terms of the maps h and the boundary operator in the manner indicated. The diagram becomes commutative "modulo homotopy equivalence classes of maps".
signaldoc is offline  
 
February 27th, 2009, 01:32 PM   #2
Member
 
Joined: Jan 2009

Posts: 72
Thanks: 0

Re: Chain Homotopy Perplexity

Quote:
Originally Posted by signaldoc
Question:

Where does the minus sign come from in the definition of chain homotopy, for example as depicted in the Wikipedia entry "Homotopy category of chain complexes"?

I have trouble motivating the definition... It appears the diagram in the cited location is not intended to be commutative. Is that correct?

Any insights would be appreciated as I am trying to teach myself homology from a nice (concise) little book by Keesee. I also have Massey but his discussion isn't easy. (In fact, the book seems to fall short of Springer's usual quality in several ways, including the typesetting.)

I'm afraid I'm more of a "point set" guy. Any other sources recommended?

Thanks!

Note added in proof: As I write this I think I'm getting a clue: We say the maps f and g are chain homotopic IF the "difference" f-g can be expressed in terms of the maps h and the boundary operator in the manner indicated. The diagram becomes commutative "modulo homotopy equivalence classes of maps".
I would remove the previous post as I have gotten a little further with my understanding, but in case anyone read it and became interested, I am providing the following.

Actually, the problem is that Keesee gives the abstract definition and then derives the consequence that is of interest. Massey, although requiring more work to get to it, starts with the theorem on identity of induced homomorphisms under homotopic maps, which he proves by constructing the necessary homomorphisms. The latter have the algebraic structure that Keesee declares a-priori, without any motivation.
signaldoc is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
chain, homotopy, perplexity



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Is connectedness an homotopy invariant? HubertM Real Analysis 1 January 19th, 2014 01:19 PM
Length of the chain biit Algebra 2 March 26th, 2013 05:47 AM
straight line homotopy-explain solution rayman Real Analysis 2 February 26th, 2013 09:44 AM
Algebraic Topology - Homotopy construction Turloughmack Topology 1 February 1st, 2011 12:03 PM
Chain rule TsAmE Calculus 4 April 24th, 2010 06:54 PM





Copyright © 2019 My Math Forum. All rights reserved.