Sumanta Das


Interest:

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. Algebraic & Geometric Topology, Volume 24-8 (2024), p. 4423-4469. arXiv: 2111.11194v3.
    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.
    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.
    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.

  • $\pi_1$-injective proper maps between non-compact surfaces. To appear in Proceedings of the American Mathematical Society. arXiv:2405.09824v2.
    We classify all $\pi_1$-injective proper maps between non-compact surfaces up to proper homotopy.

PhD Thesis:

  • Here is my PhD thesis, titled Maps Between Non-compact Surfaces.