Hatcher, Algebraic Topology, Chapter 0
13. Show that any two deformation retractions and of a space onto a subspace can be joined by a continuous family of deformation retractions , , of onto , where continuity means that the map sending to is continuous.
Proof. Given a topological space and a subspace with two deformation retractions , consider the map for which
for all , .
It’s clear that the map is continuous due to the fact that it’s a straight-line continuous mapping of one continuous map to another . In order to show that the map defines a deformation retraction, note that for arbitrary , is a linear combination of two deformation retractions. In particular, this means that for arbitrary ,
- for all .
Here, not that property 1 above stems from the fact that , property 2 from the fact that , and property 3 from the fact that for all . Hence, as claimed, is a deformation retraction for arbitrary which connects to , whereby the claim is proved.
This is not correct in non-convex subspaces of R^n. You cannot do the straight line homotopy for an arbitrary topological space, dummy!
Indeed, you are correct. Thanks for taking the time to interject your thoughts.
Multiplication of elements of by a number in isn’t even defined. I don’t see how this function is supposed to work.
You surely are correct, sir. Thanks for taking the time to respond to this; I’ll have an updated solution posted as soon as I can make the time.
I appreciate your oversight.
I thought you would be interested in a solution I came up with. I must admit, this was a very non-trivial problem for me and took me a lot of thinking. Eventually I posted my solution on stackexchange:
http://math.stackexchange.com/questions/644548/two-deformation-retractions-onto-a-are-homotopic-rel-a/653616#653616
Thanks for sharing, Christopher! I’ll check that out ASAP!
Anytime. These solutions have helped me studying for the algebraic portion of an Geometry exam I need to take and thought I could give back a little by sharing. I can share you my notes of the Chapter 1 and (especially Chapter 2) stuff when I get around to it (which should be soon). Unfortunately it will mostly be hand written stuff due to time constraints.
Anything you have would be hugely appreciated. I messed around and decided to be a topologist so I’m always combining filling in holes in my background with picking up new goodness.
The proof the previous exercise of hatcher implies that image of an element with respect to an homotopy equivalance (between a function and the identity) should stay in the path-component of the element. As both maps are such maps, along the way, the image of $x$ with respect to both deformations lie in the same path component. This observation may help to solve the problem. Concerning your website. Your effort is worth to appreciate. Have a nice mathday.