Broadly, I am interested in geometric and algebraic topology.
More precisely, my research interest is in low-dimensional topology, and my research topic includes
non-zero degree (proper) maps between two (non-compact) surfaces or two (non-compact) $3$-manifolds,
mapping class groups of finite and infinite type surfaces, and
the Goldman Lie algebra.
Here are some areas I would like to explore: Hopfian manifolds, domination of manifolds, group actions on infinite-type surfaces, simple loop conjecture, Lusternik-Schnirelmann invariants (in proper homotopy theory), $3$-manifold groups.
Publication:
Strong topological rigidity of non-compact orientable surfaces. To appear in Algebraic and Geometric Topology. arXiv: 2111.11194v3.
Abstract & Slides
We show that every orientable infinite-type surface without boundary is properly rigid as a consequence of a more general result.
Namely, we prove that if a homotopy equivalence $f\colon\ \Sigma'\to \Sigma$ between any two non-compact orientable surfaces without boundary is a proper map,
then $f$ is properly homotopic to a homeomorphism $g_\text{homeo}\colon\ \Sigma'\to \Sigma$, provided $\Sigma$ is homeomorphic to neither the plane nor the punctured plane.
Thus, all non-compact orientable surfaces without boundary, except the plane and the punctured plane, are topologically rigid in a strong sense.
Surfaces of infinite-type are non-Hopfian, with Siddhartha Gadgil. Comptes Rendus Mathématique, Volume 361 (2023), p. 1349-1356. Journal Version. arXiv:2210.03395v2.
Abstract & Slides
We show that an oriented surface $\Sigma$ without boundary is of finite-type if and only if every proper self-map $f\colon\ \Sigma\to \Sigma$ of degree $\pm 1$ is homotopic to a homeomorphism.
Thus, an oriented surface without boundary is Hopfian if and only if it is of finite-type.
The Goldman bracket characterizes homeomorphisms between non-compact surfaces, with Siddhartha Gadgil and Ajay Kumar Nair. To appear in Algebraic and Geometric Topology. arXiv:2307.02769v1.
Abstract
We show that a homotopy equivalence $f\colon\ \Sigma'\to \Sigma$ ($f$ may not be proper) between two non-compact oriented surfaces without boundary is homotopic to
a homeomorphism if and only if $\iota\left(\widehat{\alpha},\widehat{\beta}\right)=\iota\left(\widehat{f\circ \alpha}, \widehat{f\circ \beta}\right)$ for any two closed curves $\alpha,\beta\colon\ \Bbb S^1\to \Sigma'$, and
an orientation preserving homeomorphism if and only if $\left[\widehat{\alpha},\widehat{\beta}\right]=\left[\widehat{f\circ \alpha}, \widehat{f\circ \beta}\right]$ for any two closed curves $\alpha,\beta\colon\ \Bbb S^1\to \Sigma'$;
provided $\Sigma$ is homeomorphic to neither the plane nor the punctured plane. Here, $\iota(-,-)$ and $[-,-]$ denote the geometric intersection number and the Goldman Lie bracket, respectively, and $\widehat\quad$ denotes the free homotopy class of a closed curve.
In particular, for any infinite-type oriented surface $\mathbf S$ without boundary, the induced map on $\text{Out}\big(\pi_1(\mathbf S)\big)$ by a homotopy equivalence $f\colon\ \mathbf S\to \mathbf S$ is in the image of the Dehn-Nielsen-Baer-Epstein map $\text{MCG}^\pm (\mathbf S)\to\text{Out}\big(\pi_1(\mathbf S)\big)$
if and only if $f$ preserves the geometric intersection number of the free homotopy classes of any two closed curves.
In preparation:
Classification of $\pi_1$-injective proper maps between non-compact surfaces, 2024.
Degree one domination of non-compact surfaces, 2024.
The center of the Goldman Lie algebra of an infinite type surface, 2024.