Homotopy extension property for CW pairs (Hatcher)

In summary: X,\mathbb R). In summary, Hatcher's Proposition states that if a complex manifold is finite dimensional then its deformation retraction is a deformation retraction of the complex onto itself.
  • #1
ForMyThunder
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I do not understand the proof of Proposition 0.16 in Allen Hatcher's book Algebraic Topology. If someone has the book, could you please clarify the part of the proof when he says "If we perform the deformation retraction of [itex]X^n\times I[/itex] onto [itex]X^n\times\{0\}\cup (X^{n-1}\cup A^n)\times I [/itex] during the t-interval [itex][1/2^{n+1},\, 1/2^n] [/itex], this infinite concatenation of homotopies is a deformation retraction of [itex]X\times I[/itex] onto [itex] X\times\{0\}\cup A\times I [/itex]." I do not understand how this follows. Thanks in advance.
 
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  • #2
ForMyThunder said:
I do not understand the proof of Proposition 0.16 in Allen Hatcher's book Algebraic Topology. If someone has the book, could you please clarify the part of the proof when he says "If we perform the deformation retraction of [itex]X^n\times I[/itex] onto [itex]X^n\times\{0\}\cup (X^{n-1}\cup A^n)\times I [/itex] during the t-interval [itex][1/2^{n+1},\, 1/2^n] [/itex], this infinite concatenation of homotopies is a deformation retraction of [itex]X\times I[/itex] onto [itex] X\times\{0\}\cup A\times I [/itex]." I do not understand how this follows. Thanks in advance.

I don't have the book but if you tll me what A^n is I will give it a shot.
 
  • #3
lavinia, the book is available for free on Hatcher's web page
 
  • #4
ForMyThunder said:
I do not understand the proof of Proposition 0.16 in Allen Hatcher's book Algebraic Topology. If someone has the book, could you please clarify the part of the proof when he says "If we perform the deformation retraction of [itex]X^n\times I[/itex] onto [itex]X^n\times\{0\}\cup (X^{n-1}\cup A^n)\times I [/itex] during the t-interval [itex][1/2^{n+1},\, 1/2^n] [/itex], this infinite concatenation of homotopies is a deformation retraction of [itex]X\times I[/itex] onto [itex] X\times\{0\}\cup A\times I [/itex]." I do not understand how this follows. Thanks in advance.

I think the Idea is that the deformation of D[itex]^{n}[/itex] x I

onto D[itex]^{n}[/itex]x0 U D[itex]^{n-1}[/itex] X I can be followed by the cell attaching map. Over all of the n-cells this deforms X[itex]^{n}[/itex] onto X[itex]^{n}[/itex] X 0 U X[itex]^{n-1}[/itex] X I.

One then does the same thing on the remaining n-1 cells in X[itex]^{n-1}[/itex] X I and so on until you are only left with X x 0. If the complex if finite dimensional this process will stop after finitely many steps but will also work for infinite dimensional complexes such as RP[itex]^{\infty}[/itex]
 
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  • #5


The homotopy extension property for CW pairs is a fundamental property in algebraic topology that allows us to extend a homotopy defined on a subcomplex of a CW complex to the entire complex. This is a powerful tool that is used in many proofs and constructions in the field.

In Proposition 0.16 of Hatcher's book, he is proving that a CW pair (X,A) satisfies the homotopy extension property. The proof involves constructing a deformation retraction of X\times I onto X\times\{0\}\cup A\times I, which intuitively means that we can continuously deform X\times I onto X\times\{0\}\cup A\times I while keeping the points in A fixed.

The part of the proof that you are asking about involves showing that this deformation retraction can be achieved by a series of smaller deformations, each of which is defined on a smaller time interval. This is where the t-interval [1/2^{n+1},\, 1/2^n] comes into play.

Essentially, what Hatcher is saying is that if we start with a deformation retraction of X^n\times I onto X^n\times\{0\}\cup (X^{n-1}\cup A^n)\times I, and then continuously deform this onto X^{n-1}\times I, and then onto X^{n-2}\times I, and so on until we reach X\times I, we will end up with a deformation retraction of X\times I onto X\times\{0\}\cup A\times I. This is because each step in this process is a deformation retraction, and the composition of deformation retractions is also a deformation retraction.

I hope this helps clarify how the proof follows. If you are still having trouble understanding, I would suggest going through the proof step by step and trying to visualize each step in terms of the deformation retraction. Also, feel free to reach out to me for further clarification.
 

Related to Homotopy extension property for CW pairs (Hatcher)

1. What is the Homotopy extension property for CW pairs?

The Homotopy extension property for CW pairs is a property of a topological space that allows for the extension of certain types of homotopies from a subspace to the entire space. In other words, it states that if two maps are homotopic on a subspace, then they can be extended to be homotopic on the entire space.

2. What is a CW complex?

A CW complex is a type of topological space that is built up from simpler pieces called cells. These cells are attached together in a specific way, and the resulting space has a nice combinatorial structure that makes it easier to study topological properties.

3. How does the Homotopy extension property relate to CW complexes?

The Homotopy extension property is a fundamental property of CW complexes. In fact, CW complexes are defined in such a way that they automatically have the Homotopy extension property. This property is important because it allows for the study of homotopy invariants, which are important tools in algebraic topology.

4. Can you give an example of a space with the Homotopy extension property?

One example of a space with the Homotopy extension property is the n-dimensional sphere, denoted as S^n. This means that any homotopy defined on a subspace of the n-sphere can be extended to a homotopy on the entire space.

5. Are there any practical applications of the Homotopy extension property?

Yes, the Homotopy extension property has numerous practical applications in various fields such as physics, engineering, and computer science. For example, it is used in the study of topological quantum field theories and in algorithms for computing homotopy groups of spaces.

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