My Math Forum Chain Homotopy Perplexity

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 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".
February 27th, 2009, 01:32 PM   #2
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Re: Chain Homotopy Perplexity

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 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.

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