You are Here: Home >< Maths

# Homotopy Theory watch

1. Let X be a space with base point and let be the identification space obtained from the disjoint union by identifying . Prove X is a deformation retract of Y i.e. that the the inclusion is a homotopy equivalence.

so as far as i can tell, all i need to do is find a map such that and . is this correct?

well clearly
so let's take

then will be homotopic to the identity on X by the homotopy

and will be homotopic to the identity on Y by the homotopy

this seems to work to me but i'm pretty sure i'm missing some crucial point because i haven't really made use of the equivalence relation in any way whatsoever.

thanks for any help.
2. What do you mean by "xt", and so on? As far as I can tell, your space doesn't have a multiplication defined.

I haven't looked very closely; there may be other errors.
3. i don't understand your concern, either i'm being a moron or surely we just define x*t=xt
4. (Original post by latentcorpse)
Let X be a space with base point and let be the identification space obtained from the disjoint union by identifying . Prove X is a deformation retract of Y i.e. that the the inclusion is a homotopy equivalence.

so as far as i can tell, all i need to do is find a map such that and . is this correct?

well clearly
so let's take

then will be homotopic to the identity on X by the homotopy

and will be homotopic to the identity on Y by the homotopy

this seems to work to me but i'm pretty sure i'm missing some crucial point because i haven't really made use of the equivalence relation in any way whatsoever.

thanks for any help.
Helps if you just think about it in pictures. Y is just X with a copy of I glued on at x0. So just shrink I down. f:X->Y by f(x)=x is fine. Take g:Y->X to be g(y)=y if y is in X and x0 if y is in I.

As you've defined it, gf:X->X (why do you compose maps backwards?) is not homotopic to the identity on X because X need not be contractible. h makes no sense because multiplication by reals isn't necessarily defined on X. And even if it was, if X wasn't connected then h wouldn't map into X. Ditto with Y.
5. (Original post by latentcorpse)
i don't understand your concern, either i'm being a moron or surely we just define x*t=xt
What does "xt" mean?

To take a silly example: suppose X is the power set of all elements in the Periodic Table (with the discrete topology). What's x*t when x = {sodium, nitrogen}?

You're assuming X is something nice like a vector space. It might not be.
6. so are you saying i can't use xt or that i'm on the right lines but i just need to define it first?
7. using the hint in post 4 though:

i get

and also

is this right so far? i'm not quite sure how to make a homotopy between fg and the identity on y. would the homotopy have two cases as well? something like

this then raises the issue of defining yt and im not sure how to do that?
note that i have defined the identity on Y as
8. (Original post by latentcorpse)
using the hint in post 4 though:

i get

and also

is this right so far? i'm not quite sure how to make a homotopy between fg and the identity on y. would the homotopy have two cases as well? something like

this then raises the issue of defining yt and im not sure how to do that?
note that i have defined the identity on Y as
fg is already the identity on X so the h doesn't need to do anything to points of X. On [0,1], you can multiply by t here. So just set h(y, t) = ty. So, h(y,0)=y on X and 0=x0 on I. h(y, 1) = y for all Y.

Box.
9. thanks for the help but i don't understand this last statement

h(y,0)=0*y=0 surely and 0 is in I but 0=x_0 and x_0 is in X. is this what you meant?
i agree though that h(y,1)=y
10. h(y, 0) =

y for y in X

0 ~ x0 for y in I.

So h(y, 0) = fg(y)

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: February 12, 2010
Today on TSR

### TSR Pub Quiz 2018 - Anime

The first of our week-long series of entertainment quizzes

### University open days

Wed, 21 Nov '18
• Buckinghamshire New University
Wed, 21 Nov '18
• Heriot-Watt University
Wed, 21 Nov '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams