math

A note on PL handlebodies, the Hausmann trick and some homology spheres

Karim AdiprasitoLeave a comment

There is a surprising lack of intuition for PL manifolds around, which always surprised me. And it turns out you can answer some questions. We stay in the category of PL manifolds throughout. The following question was asked by Gil Kalai to Ed Swartz some 15 years ago. Ed could not answer the question, and popularized here.

Gil

Question (Kalai) Are there different (homeomorphism types of) triangulated closed compact homology (d-1)-spheres $H$ with g_3=0 (for any d)?

Without going too deep into what g_3=0 means, we note quite simply that it is equivalent to saying that H is boundary to a triangulated manifold D that has no interior simplices of dimension d-3.

The answer to this is yes. In fact, there are infinitely many (). For this purpose, we will make the following observation:

Theorem H. Let d \ge 2. Let M be a PL d-manifold. The following are equivalent:

1. M admits a PL handle decomposition into handles of index \le k,

2. M admits a PL triangulation in which all (d-k-1)-simplices are on \partial M.

Continue reading →

Sofia, Lefschetz via shelling and bonk and the holy trinity (kind of) (part 1)

Karim Adiprasito1 Comment

This is born out of an attempt to find equivariant Lefschetz elements, so to have a combinatorial Lefschetz theorem that is a little less generic for some conjectures in geometric topology. Actually this succeeds to give some interesting results, and I will update in a second part. It will take me some days, in the meantime I made this here simple and clear.

Dear X,

so, Covid seems almost over (fingers crossed). I lost three I loved during the time (none of them due to covid, funny enough; two suicides and a heart failure), I had covid twice (fingers crossed for the hat trick) despite three vaccinations. I am in Sofia, at a inaugural conference of the ICMS.

But that’s not what this post is about. This post is about another proof of the g-conjecture. Well, the Lefschetz property for simplicial cycles, really. It is the simplest one yet, but that is not why it is important. It is also the third one (that is essentially different) and combines the ideas of three teams. (I count the original one by me here, and the characteristic two proof by Papadakis and Petrotou; our joint paper is a combination of the ideas from the former and yada yada yada creative counting to make the holy number work out. Deal with it.) If the first is a refined choreography of slashes and parries that is difficult to follow, and the second the equivalent of wooshing around with bloodhound step (using a miraculous formula that comes out of nowhere), then this is the equivalent of bonking the boss with a hammer: we write down a rational function and examine it, observing it has a pole to show it is nontrivial. Unfortunately it seems to be less general than either of the previous proofs, but I will see whether it can be pushed.

Continue reading →

Degrees, curious identities, Lefschetz maps and biased pairings (and a cliffhanger)

Karim AdiprasitoLeave a comment

This post is long overdue, but being in the middle of nowhere helps finally progressing a little.

In the last post, I discussed a lemma of Kronecker. Let me discuss another version

Lemma (Kronecker, second version) Consider A, B linear maps from a vectorspace X to another one Y over any infinite field. Assume that im(A) and B ker(A) intersect trivially. Then the generic linear combination of A and B has kernel equal to the intersection of kernels of A and B, in other words, the kernel is as small as possible.

This is a very powerful tool to construct high-rank maps in vector spaces, in particular in the case when X and Y are dual in a Poincaré duality algebra. Then im A and ker A are orthogonal complements, and linear algebra tells us that orthogonal complements intersect trivially if and only if the Poincaré pairing does not degenerate on either space. This is exploited in my proof of the g-conjecture to construct Lefschetz elements, see here, and also in this survey (which I was invited to for winning the EMS prize, which I still cannot quite believe).

Continue reading →

Schmidt and analytic ranks, the Kronecker lemma and Hadamard lectures

Karim Adiprasito2 Comments

Just a short post: David Kazhdan, Tammy Ziegler and I have recently (and finally) posted our proof that the analytic rank of a cubic, the logarithm of its bias, and its Schmidt rank (how hard it is to write it as products of lower degree polynomials), are linearly related.

Coauthors Tammy and David

This came out of our discussion of Kronecker’s lemma. One version can be stated as follows:

Continue reading →

Minkowski sums (and how big they must be)

Karim AdiprasitoLeave a comment

So, this is the promised actual first post, motivated by and answering a question recently posed to me by Yue Ren (and several before him that I apologize for forgetting). It comes with a problem, and that is, to prove the result in a simpler, more elementary way.

I had written a longer paper (with Raman Sanyal) on a problem concerning Upper Bounds on the complexity of Minkowski sums that dealt with the problem quite thoroughly.

Yue Ren (who asked me most recently), and Raman Sanyal (coauthor)

But after we submitted the paper, and it was accepted and published in Publications IHES, someone asked me a question. I thought for a while, and answered that it follows from that and that lemma in the paper. But the question came up again and again, seemingly being quite relevant. On the other hand, writing a new paper on the matter was tedious, as it promised to be 90% repetition of paper number one. Which was an annoying prospect at best.

The problem is this: Consider a bunch (say m of polytopes (P_i) in a euclidean vector space of dimension d. For simplicity, and to be interesting, let us assume that all these polytopes are of positive dimension. Also, let us assume that all these polytopes are in general position with respect to each other.

Question: How many vertices must the Minkowski sum of the family (P_i) have? Surely, it has to be at least the number of vertices as the Minkowski sum of m segments in general position.

Continue reading →