arXiv subjects
Publications and source records attributed to Ajay Singh Thakur.
Let G be a finite group. We study the group of G-equivariant self-homotopy equivalences of product of G-spaces. For a product of n-spaces, we represent it as product of n-subgroups under the assumption of equivariant reducibility. Further we describe each factor as a split short exact sequence. Also, we obtain an another kind of factorisation, called $LU$ type decomposition, as product of two subgroups.
We give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others. It also yields large classes of new examples of non-Kähler compact complex manifolds. Moreover, under suitable restrictions on the base manifold, the structure group, and characteristic classes, the total space of the principal bundle admits SKT metrics. This generalizes recent results of Grantcharov et al. We study the Picard group and the algebraic dimension of the total space in some cases. We also use a slightly generalized version of the construction to obtain (non-Kähler) complex structures on tangential frame bundles of complex orbifolds.
In this article we show that there are at most two integers up to $2(n-k)$, which can occur as the degrees of nonzero Stiefel-Whitney classes of vector bundles over the Stiefel manifold $V_k(\mathbb{R}^n)$. In the case when $n> k(k+4)/4$, we show that if $w_{2^q}(ξ)$ is the first nonzero Stiefel-Whitney class of a vector bundle $ξ$ over $V_k(\mathbb{R}^n)$ then $w_t(ξ)$ is zero if $t$ is not a multiple of $2^q.$ In addition, we give relations among Stiefel-Whitney classes whose degrees are multiples of $2^q$.
In this note we give a necessary condition for having an almost complex structure on the product $S^{2m} \times M$, where $M$ is a connected orientable closed manifold. We show that if the Euler characteristic $χ(M) \neq 0$, then except for finitely many values of $m$, we do not have almost complex structure on $S^{2m} \times M$. In the particular case when $M = \mathbb{C}\mathbb P^n, n \neq 1$, we show that if $n \not \equiv 3 \pmod 4$ then $S^{2m} \times \mathbb C \mathbb P^{n}$ has an almost complex structure if and only if $m = 1,3$. As an application we obtain conditions on the nonexistence of almost complex structure on Dold manifolds.
A finite $CW$-complex $X$ is $C$-trivial if for every complex vector bundle $ξ$ over $X$, the total Chern class $c(ξ)=1$. In this note we completely determine when each of the following spaces are $C$-trivial: suspensions of stunted real projective spaces, suspensions of stunted complex projective spaces and suspensions of stunted quaternionic projective spaces.
Let $X^k_{m,n}=Σ^k (\mathbb R\mathbb P^m/\mathbb R\mathbb P^n)$. In this note we completely determine the values of $k,m,n$ for which the total Stiefel-Whitney class $w(ξ)=1$ for any vector bundle $ξ$ over $X^k_{m,n}$.
A space $X$ is called $W$-trivial if for every vector bundle $ξ$ over $X$, the total Stiefel-Whitney class $W(ξ)= 1$. In this article we shall investigate whether the suspensions of Dold manifolds, $\s^k D(m,n)$, is $W$-trivial or not.
We define the notion of characteristic rank, $\mathrm{charrank}_X(ξ)$, of a real vector bundle $ξ$ over a connected finite $CW$-complex $X$. This is a bundle-dependent version of the notion of characteristic rank introduced by Július Korbaš in 2010. We obtain bounds for the cup length of manifolds in terms of the characteristic rank of vector bundles generalizing a theorem of Korbaš and compute the characteristic rank of vector bundles over the Dold manifolds, the Moore spaces and the stunted projective spaces amongst others.
The characteristic rank of a vector bundle $ξ$ over a finite connected $CW$-complex $X$ is by definition the largest integer $k$, $0\leq k\leq \mathrm{dim}(X)$, such that every cohomology class $x\in H^j(X;\mathbb Z_2)$, $0\leq j\leq k$, is a polynomial in the Stiefel-Whitney classes $w_i(ξ)$. In this note we compute the characteristic rank of vector bundles over the Stiefel manifold $V_k(\mathbb F^n)$, $\mathbb F=\mathbb R,\mathbb C,\mathbb H$.
Let $\bar{L}_i\lr X_i$ be a holomorphic line bundle over a compact complex manifold for $i=1,2$. Let $S_i$ denote the associated principal circle-bundle with respect to some hermitian inner product on $\bar{L}_i$. We construct complex structures on $S=S_1\times S_2$ which we refer to as {\em scalar, diagonal, and linear types}. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that $\bar{L}_i$ are equivariant $(\bc^*)^{n_i}$-bundles satisfying some additional conditions. The linear type complex structures are constructed assuming $X_i$ are (generalized) flag varieties and $\bar{L}_i$ negative ample line bundles over $X_i$. When $H^1(X_1;\br)=0$ and $c_1(\bar{L}_1)\in H^2(X_1;\br)$ is non-zero, the compact manifold $S$ does not admit any symplectic structure and hence it is non-Kähler with respect to {\em any} complex structure. We obtain a vanishing theorem for $H^q(S;\mathcal{O}_S)$ when $X_i$ are projective manifolds, $\bar{L}_i^\vee$ are very ample and the cone over $X_i$ with respect to the projective imbedding defined by $\bar{L}_i^\vee$ are Cohen-Macaulay. We obtain applications to the Picard group of $S$. When $X_i=G_i/P_i$ where $P_i$ are maximal parabolic subgroups and $S$ is endowed with linear type complex structure with `vanishing unipotent part' we show that the field of meromorphic functions on $S$ is purely transcendental over $\bc$.