I am broadly interested in geometric topology and algebraic topology.
So far, my research output falls under the following headings: maps of non-zero degree between low-dimensional manifolds,
mapping class groups of finite and infinite type surfaces, and the Goldman Lie algebra.
Recently, I have started working on the obstruction to the existence of almost complex structures.
Publication:
Strong topological rigidity of non-compact orientable surfaces. To appear in Algebraic & 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, 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. 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 & Geometric Topology. arXiv:2307.02769v2.
Abstract & Slides
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.
Pre-print:
Classification of $\pi_1$-injective proper maps between non-compact surfaces. arXiv:2405.09824v1.
Abstract
We show that almost all $\pi_1$-injective proper maps between two non-compact orientable surfaces without boundary, where surfaces are possibly of infinite type, can be properly homotoped to finite-sheeted covering maps.
More precisely, we prove the following three theorems.
Theorem 1.1 Let $\Sigma',\Sigma$ be two non-compact oriented surfaces without boundary such that $\Sigma'$ is neither the plane nor the punctured plane.
Suppose $f\colon \Sigma'\to \Sigma$ is a $\pi_1$-injective proper map.
Then $f$ is properly homotopic to a $d$-sheeted covering map $p\colon \Sigma'\to \Sigma$ for some positive integer $d$.
Thus, $\deg(f)=\pm d(\neq 0)$. Moreover, $\Sigma'$ is of infinite type if and only if $\Sigma$ is of infinite type.
Theorem 1.2 Suppose $f$ is a $\pi_1$-injective proper map from $\Bbb C^*:=\Bbb C\setminus \{0\}$ to a non-compact oriented surface $\Sigma$ without boundary. Let $n:= \deg(f)$. Then, we have the following.
If $n=0$, then there exists a $\pi_1$-injective, proper embedding $\iota\colon \Bbb S^1\times [0,\infty)\hookrightarrow \Sigma$,
along with a non-zero integer $d$, such that after a proper homotopy, $f$ can be described by the proper map $\Bbb S^1\times \Bbb R\ni (z,t)\longmapsto \iota\left(z^d,|t|\right)\in \Sigma$.
Thus, $\Sigma$ has an isolated planar end, and given any compact subset $K$ of $\Sigma$, there exists a proper map $g$ properly homotopic to $f$ such that $\text{im}(g)\subseteq \Sigma\setminus K$.
If $n\neq 0$, then $\Sigma=\Bbb C^*$ and $f$ is properly homotopic to the covering $\Bbb C^*\ni z\longmapsto z^n\in \Bbb C^*$ if $n>0$,
and to the covering $\Bbb C^*\ni z\longmapsto \overline z^{-n}\in \Bbb C^*$ if $n<0$.
Theorem 1.3 Suppose $f$ is a proper map from $\Bbb C$ to a non-compact oriented surface $\Sigma$ without boundary. Let $n:=\deg(f)$. Then, we have the following.
If $n=0$, then for every compact subset $K$ of $\Sigma$, there exists a proper map $g$ properly homotopic to $f$ such that $\text{im}(g)\subseteq \Sigma\setminus K$.
If $n\neq 0$, then $\Sigma=\Bbb C$, and $f$ is properly homotopic to the branched covering $\Bbb C\ni z\longmapsto z^n\in \Bbb C$ if $n>0$,
and to the branched covering $\Bbb C\ni z\longmapsto \overline z^{-n}\in \Bbb C$ if $n<0$.
PhD Thesis:
Here is my PhD thesis, titled Maps Between Non-compact Surfaces.