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Homotopy Equivalent watch

1. so to show h is well defined i only need to show that

where i used that under the equivalence relation.
2. (Original post by SsEe)
h is showing the homotopy equivalence of two maps (namely and . So it should be a map
so shouldn't we use
as this maps from to thus defining a homotopy between the two maps as required.

but then
3. (Original post by latentcorpse)
so to show h is well defined i only need to show that

where i used that under the equivalence relation.
Yeah that's fine.

so shouldn't we use
as this maps from to thus defining a homotopy between the two maps as required.

but then
Back to basics as you're getting confused about what's mapping where. Look at the definition of two maps being homotopic. The situation here is that we have well defined maps:

and

and we want a homotopy h between them. By definition, that's a map

such that and
That's what I defined a few posts ago.
4. (Original post by SsEe)
Then you define the homotopy
by . This is again well defined under the identification (exercise for reader...)
your homotopy is defiend for maps between and though. Doesn't it need altered?

should it be

?
5. (Original post by latentcorpse)
your homotopy is defiend for maps between and though. Doesn't it need altered?

should it be

?
But is with an identification on the top and bottom edge. I define r and h on as the notation is easier. As you said, a parametrization for the moebius band isn't obvious. Then, I show that r and h are well defined with the identification and hence give well defined maps on

It is a slight abuse of notation, but it's a "standard" abuse of notation I think! If you like, you can write for the homotopy under the identifcation.
6. (Original post by SsEe)
It is a slight abuse of notation, but it's a "standard" abuse of notation I think! If you like, you can write for the homotopy under the identifcation.
so i'd write out the and say that this means there is a homotopy
then i show that h is well deifned under the identification and so that leads to a homotopy
i.e. a homotopy

is all the notation ok above?

also i defined differently in the two cases (first i said it was from and then from

finally i said "leads to" . surely we need to be mroe precise, no?

thanks again. i'm probably being a bit pedantic now.
7. (Original post by latentcorpse)
so i'd write out the and say that this means there is a homotopy
then i show that h is well deifned under the identification and so that leads to a homotopy
i.e. a homotopy

is all the notation ok above?

also i defined differently in the two cases (first i said it was from and then from

finally i said "leads to" . surely we need to be mroe precise, no?

thanks again. i'm probably being a bit pedantic now.
You probably want hats on after you've identified the edges.

I'd probably say gives rise to a well defined map on the quotient space. It is all a bit pedantic. Another mathematician would understand it. Even without the hat notation.
8. (Original post by SsEe)
You probably want hats on after you've identified the edges.

I'd probably say gives rise to a well defined map on the quotient space. It is all a bit pedantic. Another mathematician would understand it. Even without the hat notation.
hang on a second here. i just realised that i was meant to use functions from and as described in the question and as we did in the first part.

what about
and
for

then we get

and so is obviously homotopic to the identity on
but also

and again we are required to prove is homotopic to
we define
which is a continuous map from to and therefore defines a homotopy

how's that look? it seems simpler than not using and having to use the deformation retract.
9. (Original post by latentcorpse)
hang on a second here. i just realised that i was meant to use functions from and as described in the question and as we did in the first part.

what about
and
for

then we get

and so is obviously homotopic to the identity on
but also

and again we are required to prove is homotopic to
we define
which is a continuous map from to and therefore defines a homotopy

how's that look? it seems simpler than not using and having to use the deformation retract.
It's exactly the same map, just with a different parametrization. You still need to check it's well defined at the join. This question was done and dusted many posts ago!

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