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##### Activity
• Jul 19 2020 10:05

• Jul 19 2020 10:05
• Jul 19 2020 09:25
bollu commented #2
• Jul 19 2020 09:24
bollu synchronize #2
• Jul 19 2020 08:35
• Jul 19 2020 08:34
• Jul 19 2020 07:10
bollu opened #2
• Apr 02 2020 05:40

• Mar 31 2020 07:17

poincare duality notes notes reformatted, pictures (compare)

• Mar 30 2020 10:12

note sketch for duality lecture. (compare)

• Mar 28 2020 10:59

• Mar 27 2020 14:44

notes on free resolutions start of lecture draft (compare)

• Mar 26 2020 07:00

• Mar 26 2020 06:56

• Mar 25 2020 11:28

first online lecture made notes (compare)

• Mar 24 2020 04:05

tree homotopy picture Merge branch 'master' of https:… (compare)

• Mar 24 2020 01:11

• Mar 24 2020 01:11
• Mar 24 2020 01:08
Chinmaya Kausik
@Chinmaya-Kausik
To achieve the formal linear combination picture.
That's what it looks like. Indeed one converts a rectangle into a simplex in two different ways for the two different associations. Once this is done, the two coincide.
Chinmaya Kausik
@Chinmaya-Kausik
I see, thank you so much.
Thank You. I too understand this clearly now.
If any of you is online can you go to teams.microsoft.com and log in with your iisc account. I want to test this as an alternative, since we have a paid subscription.
Chinmaya Kausik
@Chinmaya-Kausik
As you may have seen on teams.
AR-MA210
@AR-MA210
I saw another version of Whitehead's theorem somewhere, for homology groups, stated as "If a continuous map between simply connected CW complexes $f:X \to Y$ induces isomorphisms $H_*(X) \cong H_*(Y)$, $f$ is a homotopy equivalence.". Is it possible to prove this using the theorem from today's class + Hurewicz's theorem or some other tools we have?
It is another theorem.
It is in Hatcher. It follows from the result that I stated and did not prove, plus Hurewicz theorem and a little more
AR-MA210
@AR-MA210
Ok, I'll look into it. Thanks
I have started the teams, but not yet the video.
Sumanta
@dasumanta
Okay.
When I went to the link, I found they even have a debian linux app (they will surely have windows and mac), so I downloaded this. It may be good for everyone to do the same.
Shabarish CH
I am on the app for MacOS
philipthomasarun
@philipthomasarun
I am on the app on Windows
Chinmaya Kausik
@Chinmaya-Kausik
I'm trying to launch it on Chrome
Shabarish CH
It told me to restart the app to join the meeting
Chinmaya Kausik
@Chinmaya-Kausik
I had a question for Prof. Gadgil ( @siddhartha-gadgil ). We have seen lens spaces, but I think we're not sure why studying them is useful/relevant/interesting. I thought you would have some important insights on this, also because it seems you have worked on them in the past.
There are many interesting things with them. First of all, they are the historically first, and still prototypical examples for homotopy equivalence not coiniding with homeomorphism.
To show this involves both homotopy classification and homeomorphism classification. Both involve interesting ideas.
They are also geometric manifolds.
Further, Milnor's "Two complexes that are homeomorphic but combinatorially distinct" are derived from lens spaces.
Chinmaya Kausik
@Chinmaya-Kausik

To show this involves both homotopy classification and homeomorphism classification. Both involve interesting ideas.

I see. Is there some sort of punchline here? Or are the ideas themselves somewhat non-trivial?

I would say more than one punchline.
Let me stick to $L(p, q)$ with $p$ prime. Firstly, any isomorphism of the fundamental group is induced by a map, but its degree $(mod p)$ is determined by the isomorphism.
Hence whether this is a homotopy equivalence, and further orientation preserving, can be determined.
I can explain these briefly in one of our meetings, after the core material.
On the other hand, there is more than one proof of homeomorphism classification, with different proofs generalizing in different directions and illustrating different things.
Incidentally, we can show that the left and right trefoils are different using the oriented homotopy classification of lens spaces.
Chinmaya Kausik
@Chinmaya-Kausik

Incidentally, we can show that the left and right trefoils are different using the oriented homotopy classification of lens spaces.

Oh, I see. Very interesting. I suppose this comes from them being torus knots and then using the solid torus decomposition of $S^3$?

And perhaps using $\mathbb{Z}_3$ as homeomorphisms on the torus for quotienting to get a lens space
No, we take the 2-fold branched cover over the knot. The two trefoils give $L(3, 1)$ and $L(3, -1) = L(3, 2)$. There is no orientation preserving homotopy equivalence between these.
And yes, it is related to them being torus knots.
Chinmaya Kausik
@Chinmaya-Kausik

I can explain these briefly in one of our meetings, after the core material.

Yes, that would be nice, I guess. At least from my side.

Chinmaya Kausik
@Chinmaya-Kausik
When is the next class, by the way? Mentioning Prof. Gadgil ( @siddhartha-gadgil )
Tuesday 9:30 if it is fine with everyone. Will email with this and some more details Monday morning.
Chinmaya Kausik
@Chinmaya-Kausik
Fine by me
ajay-k-nair
@ajay-k-nair
Fine by me!
Sumanta
@dasumanta
Fine.
Shabarish CH