Current Session

About the speaker : 

Pravin is a graduate student at IISER Mohali working under Dr. Mahendra Singh. His research interest lies in knot theory, Coxeter groups, Artin groups, and mapping class groups of surfaces. 

In this seminar, Sumanta will talk about a joint work with Prof. Gadgil where they have shown that finite-type surfaces are characterized by a topological analogue of the Hopf property. Namely, an orientable surface Σ is of finite-type if and only if every proper map f : Σ → Σ of degreeone is homotopic to a homeomorphism.

Given an orientable, compact, and connected surface S_g with genus g > 0, we define the mapping class group as isotopy classes of self-homeomorphisms of S_g. Dehn-Lickorish Theorem, Nielsen-Thurston Classification theorem, and Dehn-Nielsen-Baer theorem are three of the most important results in the surface theory. In this talk, we would try to motivate these three results by looking at the case when g = 1, i.e., the torus. In the case of torus, these results are easier to see. If time permits, I would try to give a basic sketch of the proofs of at least one of the theorems.

For g ≥ 2, let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g. In this talk, we will discuss a complete characterization of the infinite metacyclic subgroups of Mod(Sg) up to conjugacy. In particular, we discuss equivalent conditions under which a pseudo-Anosov mapping class generates a metacyclic subgroup of Mod(Sg) with another mapping class. As an application to our main results, we show the existence of certain infinite metacyclic subgroups of Mod(Sg) explicitly. If time permits, we derive bounds on the order of a periodic generator of an infinite metacyclic subgroup of Mod(Sg).

In 1960s, Lickorish and Wallace (independently) proved that any closed, orientable smooth 3-manifolds can be obtained by performing an integral surgery on a link in S3. Since the conserve is trivial, we get a correspondence between framed links in S3 and closed, oriented smooth 3-manifold. But this correspondence is not one-to-one, we may have two framed links corresponding to the same 3-manifold.

In 1970, Kirby proved that given such a pair of framed links there is a finite sequence of moves to obtain one link from another. These moves are called Kirby Moves, and there are two types of such moves. In this, I will talk about the Kirby moves. We will see some examples of these moves in action. If time permits, we may talk about the contact geometric analogs of the Lickorish-Wallace theorem, and Kirby moves.