arXiv subjects
Publications and source records attributed to Somnath Basu.
We prove a result analogous to Reeb's theorem in the context of Morse-Bott functions: if a closed, smooth manifold $M$ admits a Morse-Bott function having two critical submanifolds $S^k$ and $S^l$ ($k \neq l$), then $M$ has dimension $k+l+1$ and is homeomorphic to the standard sphere $S^{k+l+1}$ but not necessarily diffeomorphic to it. We also prove similar results for projective spaces over the real numbers, complex numbers and quaternions.
Given a fibration $F \hookrightarrow E \rightarrow B$ with a homotopy section $s: B \rightarrow E$, James introduced a binary product $\left\{, \right\}_s: \pi_i B \times \pi_j F \rightarrow \pi_{i+j-1} F$, called the brace product, which was later generalized by Yoon. We show that the vanishing of this generalized brace product is the precise obstruction to the $H$-splitting of the loop space fibration, i.e., $\Omega E \simeq \Omega B \times \Omega F$ as $H$-spaces. Using rational homotopy theory, we show that for rational spaces, the vanishing of the generalized brace product coincides with the vanishing of the classical James brace product, enabling us to perform the relevant computations. In addition, the notion of $J$-homomorphism is generalized and connected to the generalized brace product. Among the applications, we characterize the homotopy types of certain fibrations, including sphere bundles over spheres.
We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in physics. We also analyze a variant of these polynomials for colored embedded graphs. This is used to describe the change in the polynomial under basic graph theoretic operations. We conclude with several applications of this polynomial including detection of certain classes of graphs and the connection of this polynomial with topological entanglement entropy.
We present a combinatorial isomorphism between Stasheff associahedra and an inductive cone construction of those complexes given by Loday. We give an alternate description of certain polytopes, known as multiplihedra, which arise in the study of $A_\infty$ maps. We also prove a combinatorial isomorphism between Stasheff associahedra, collapsed multiplihedra and graph cubeahedra for path graphs.
It is well known that the topological entanglement entropy ($S_{topo}$) of a topologically ordered ground state in 2 spatial dimensions can be captured efficiently by measuring the tripartite quantum information ($I^{3}$) of a specific annular arrangement of three subsystems. However, the nature of the general N-partite information ($I^{N}$) and quantum correlation of a topologically ordered ground state remains unknown. In this work, we study such $I^N$ measure and its nontrivial dependence on the arrangement of $N$ subsystems. For the collection of subsystems (CSS) forming a closed annular structure, the $I^{N}$ measure ($N\geq 3$) is a topological invariant equal to the product of $S_{topo}$ and the Euler characteristic of the CSS embedded on a planar manifold, $|I^{N}|=χS_{topo}$. Importantly, we establish that $I^{N}$ is robust against several deformations of the annular CSS, such as the addition of holes within individual subsystems and handles between nearest-neighbour subsystems. For a general CSS with multiple holes ($n_{h}>1$), we find that the sum of the distinct, multipartite informations measured on the annular CSS around those holes is given by the product of $S_{topo}$, $χ$ and $n_{h}$, $\sum^{n_{h}}_{μ_{i}=1}|I^{N_{μ_{i}}}_{μ_{i}}| = n_{h}χS_{topo}$. The $N^{th}$ order irreducible quantum correlations for an annular CSS of $N$ subsystems is also found to be bounded from above by $|I^{N}|$, which shows the presence of correlations among subsystems arranged in the form of closed loops of all sizes. Our results offer important insight into the nature of the many-particle entanglement and correlations within a topologically ordered state of matter.
Associated to every closed, embedded submanifold $N$ in a connected Riemannian manifold $M$, there is the distance function $d_N$ which measures the distance of a point in $M$ from $N$. We analyze the square of this function and show that it is Morse-Bott on the complement of the cut locus $\mathrm{Cu}(N)$ of $N$, provided $M$ is complete. Moreover, the gradient flow lines provide a deformation retraction of $M-\mathrm{Cu}(N)$ to $N$. If $M$ is a closed manifold, then we prove that the Thom space of the normal bundle of $N$ is homeomorphic to $M/\mathrm{Cu}(N)$. We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group $U(p,q)$ to $U(p)\times U(q)$ and a geometric deformation of $GL(n,\mathbb{R})$ to $O(n,\mathbb{R})$ which is different from the Gram-Schmidt retraction.
In this paper, we develop a systematic approach to enumerate curves with a certain number of nodes and one further singularity which maybe more degenerate. As a result, we obtain an explicit formula for the number of curves in a sufficiently ample linear system, passing through the right number of generic points, that have $δ$ nodes and one singularity of codimension $k$, for all $δ+k \leq 8$. In particular, we recover the formulas for curves with upto six nodal points obtained by Vainsencher. Moreover, all the codimension seven numbers we have obtained agree with the formulas obtained by Kazarian. Finally, in codimension eight, we recover the formula of A.Weber, M.Mikosz and P.Pragacz for curves with one singular point and we also recover the formula of Kleiman and Piene for eight nodal curves. All the other codimension eight numbers we have obtained are new.
This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu-Poisson manifold (of order $n>2$) gives rise to a special bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid (of order $n$). It is further demonstrated that such bialgebroids canonically induce a Nambu-Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid over a point.
In this paper we study the mod $2$ cohomology ring of the Grasmannian $\widetilde{G}_{n,3}$ of oriented $3$-planes in $\mathbb{R}^n$. We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost complete description of the cohomology ring. This partial description allows us to provide lower and upper bounds on the cup length of $\widetilde{G}_{n,3}$. As another application, we show that the upper characteristic rank of $\widetilde{G}_{n,3}$ equals the characteristic rank of $\widetildeγ_{n,3}$, the oriented tautological bundle over $\widetilde{G}_{n,3}$.
For $n\geq 2$ we consider $(n-1)$-connected closed manifolds of dimension at most $(3n-2)$. We prove that away from a finite set of primes, the $p$-local homotopy groups of $M$ are determined by the dimension of the space of indecomposable elements in the cohomology ring $H^\ast(M)$. Moreover, we show that these $p$-local homotopy groups can be expressed as direct sum of $p$-local homotopy groups of spheres. This generalizes some of the results of our earlier work.
Let M be a smooth manifold which is homeomorphic to the n-fold product of S^k, where k is odd. There is an induced homomorphism from the group of diffeomorphisms of M to the automorphism group of H k (M ; Z). We prove that the image of this homomorphism contains a congruence subgroup of SL_n (Z) whenever n is at least 3.
In this paper we give a formula for the homotopy groups of $(n-1)$-connected $2n$-manifolds as a direct sum of homotopy groups of spheres in the case the $n^{th}$ Betti number is larger than $1$. We demonstrate that when the $n^{th}$ Betti number is $1$ the homotopy groups might not have such a decomposition. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we establish a conjecture of J. C. Moore.
In this paper we obtain an explicit formula for the number of curves in a compact complex surface $X$ (passing through the right number of generic points), that has up to one node and one singularity of codimension $k$, provided the total codimension is at most $7$. We use a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle $V$ over $M$, counted with signs, is the Euler class of $V$ evaluated on the fundamental class of $M$.
In this paper we obtain an explicit formula for the number of curves in two dimensional complex projective space, of degree d, passing through d(d+3)/2-(k+1) generic points and having one node and one codimension k singularity, where k is at most 6. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M, counted with a sign, is the Euler class of V evaluated on the fundamental class of M.
Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We show that for a generic metric on such a smooth four manifold with second Betti number at least three, the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. The number of closed Reeb orbits of length at most l on the spherization of the cotangent bundle also grow exponentially for any Reeb flow.
We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.
In this paper we obtain an explicit formula for the number of degree d curves in two dimensional complex projective space, passing through (d(d+3)/2 -k) generic points and having a codimension k singularity, where k is at most 7. In the past, many of these numbers were computed using techniques from algebraic geometry. In this paper we use purely topological methods to count curves. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M, counted with a sign, is the Euler class of V evaluated on the fundamental class of M.
Associated to every connected, topological space $X$ there is a Hopf algebra - the Pontrjagin ring of the based loop space of the configuration space of two points in X. We prove that this Hopf algebra is not a homotopy invariant of the space. We also exhibit interesting examples of H-spaces, which are homotopy equivalent as spaces, which either lead to isomorphic rational Hopf algebras or not, depending crucially on the existence of Whitehead products. Moreover, we investigate a (naturally motivated) twisted version of these Pontrjagin rings in the various aforementioned contexts. In all of these examples, Massey products abound and play a key role.