Abstract Nonsense

Crushing one theorem at a time

Homotopy and the Homotopy Category (Pt. III)


Point of Post: This is a continuation of this post.

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The Homotopy Category

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The notion of homotopy allows us to define a new category— the homotopy category \mathbf{hTop}. The objects of \mathbf{hTop} are still topological spaces, but now the arrows X\to Y in \mathbf{hTop} are not continuous maps X\to Y but instead are homotopy classes of continuous maps X\to Y–thus, \text{Hom}_{\mathbf{hTop}}(X,Y)=[X,Y]. The composition law in \mathbf{hTop} is the obvious–representative wise composition (function composition!) of homotopy classes. In other words, if X\xrightarrow{[f]}Y\xrightarrow{[g]}Z then we define [g]\circ[f] to be [g\circ f]. We claim that this is well-defined:

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Theorem: Let f,g:X\to Y and h,k:Y\to Z be such that f\simeq g and h\simeq k, then h\circ f\simeq k\circ g.

Proof: Suppose that H is a homotopy from f to g and F is a homotopy from h to k. Define the map G:X\times[0,1]\to Z defined by G(x,t)=F(H(x,t),t). Clearly G is continuous since it is the composition of the maps F\circ R where R is the map X\times[0,1]\to Y\times[0,1] defined by R(x,t)=(H(x,t),t) (which is continuous since each coordinate function is continuous). Now, it’s easy to see that

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G(x,0)=F(H(x,0),0)=F(f(x),0)=h(f(x))=(h\circ f)(x)

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and

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G(x,1)=F(H(x,1),1)=F(g(x),1)=k(g(x))=(k\circ g)(x)

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from where it follows that G is a homotopy from h\circ f to k\circ g and so the theorem follows. \blacksquare

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Now, it’s clear that the identity arrow for each object X in \mathbf{hTop} is the homotopy class [\text{id}_X]\in[X,X]. It totally makes sense, at least with this definition, why the notion of homotopy equivalence was defined as it was. Namely, it’s clear that the notion of a homotopy equivalence f:X\to Y is precisely what it means for [f]\in [X,Y] to be an isomorphism in \mathbf{hTop}.

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References:

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[1] Spanier, Edwin Henry. Algebraic Topology. New York: McGraw-Hill, 1966. Print.

[2] Hatcher, Allen. Algebraic Topology. Cambridge: Cambridge UP, 2002. Print.

[3] Bredon, Glen E. Topology and Geometry. New York: Springer-Verlag, 1993. Print.

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August 30, 2012 - Posted by | Algebraic Topology, Topology | , , , , ,

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