arXiv subjects

Manas Mandal

Publications and source records attributed to Manas Mandal.

6 recordsLinked to original sources

arXiv2026

Degrees of Maps between Generalized Dold Manifolds

We study the existence of maps of nonzero Brouwer degree between two orientable, equal-dimensional generalized Dold manifolds (GDMs) fibered by complex partial flag manifolds over real projective spaces. The GDMs considered here are obtained as orbit spaces of the diagonal involution on the product of a sphere and a complex partial flag manifold, acting antipodally on the sphere and by complex conjugation on the flag manifold. We show that no map of nonzero degree exists between two distinct GDMs when their base real projective spaces are different, except in one exceptional case. Furthermore, when the base real projective spaces coincide but fibers are distinct, we prove the nonexistence of maps of nonzero degree in most cases, including those in which at least one of the fibers is not a Grassmannian. As an application of our study, we establish cohomological rigidity for the class of products of a sphere with a complex or quaternionic partial flag manifold.

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arXiv2026

Degrees of Maps and Cohomological Rigidity of Partial Flag Manifolds

We study the existence of continuous maps with nonzero Brouwer degree between partial flag manifolds. We prove that every continuous map between distinct complex or quaternionic partial flag manifolds has degree zero if either the domain or the codomain is not a Grassmannian. This establishes, for such partial flag manifolds, the analogue of the results of Ramani-Sankaran and Sankaran-Sarkar for complex and quaternionic Grassmannians, as well as an algebraic-geometric result of Paranjape and Srinivas for complex Grassmannians. As a consequence, we prove that complex and quaternionic partial flag manifolds are cohomologically rigid; that is, their rational cohomology rings determine their homeomorphism types. We conjecture that complex and quaternionic partial flag manifolds are homologically rigid.

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arXiv2026

On Generalized Milnor Manifolds and Their Topological Complexity

We introduce generalized Milnor manifolds (GMM), extending the classical Milnor manifolds over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$. We compute their integral cohomology algebras in the complex and quaternionic cases and their mod-$2$ cohomology algebras in the real case. We compare GMM with partial flag manifolds, investigate when they are homotopy equivalent, and obtain a necessary and sufficient condition in the complex and quaternionic cases. We further prove that complex GMM are K\"ahler manifolds. As an application, we determine their higher topological complexities, obtaining exact values in the complex and quaternionic cases and bounds in the real case. Along the way, for a fibre bundle satisfying the Leray--Hirsch hypothesis, we establish a lower bound for the zero-divisor cup-length of the total space in terms of base and fibre.

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arXiv2025

Rigidity of cohomology automorphisms of homogeneous spaces and coincidence theory

We obtain a rigidity phenomena of rational cohomology automorphisms of certain homogeneous spaces, in the presence of external cohomology classes arising from spaces with trivial cup product in rational cohomology algebra. We classify graded endomorphisms of the rational cohomology algebra of the product of a sphere and a complex Grassmannian, whose images are nonzero in the second cohomology of the Grassmannian. We also derive necessary conditions for the generalized Dold spaces to satisfy the coincidence property, in particular the fixed-point property. As an application of our results, we obtain several sufficient conditions for the existence of a point of coincidence between a pair of continuous functions on certain generalized Dold spaces.

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arXiv2024

Cohomology and K-theory of generalized Dold manifolds fibred by complex flag manifolds

Let $\nu=(n_1,\ldots, n_s), s\ge 2,$ be a sequence of positive integers and let $n=\sum_{1\le j\le s}n_j$. Let $\mathbb CG(\nu)=U(n)/(U(n_1)\times \cdots\times U(n_s))$ be the complex flag manifold. Denote by $P(m,\nu)=P(\mathbb S^m,\mathbb CG(\nu))$ the generalized Dold manifold $\mathbb S^m\times \mathbb CG(\nu)/\langle \theta\rangle $ where $\theta=\alpha\times \sigma$ with $\alpha:\mathbb S^m\to \mathbb S^m$ being the antipodal map and $\sigma:\mathbb CG(\nu)\to \mathbb CG(\nu)$, the complex conjugation. The manifold $P(m,\nu)$ has the structure of a smooth $\mathbb CG(\nu)$-bundle over the real projective space $\mathbb RP^m.$ We determine the additive structure of $H^*(P(m,\nu);R)$ when $R=\mathbb Z$ and its ring structure when $R$ is a commutative ring in which $2$ is invertible. As an application, we determine the additive structure of $K(P(m,\nu))$ almost completely and also obtain partial results on its ring structure. The results for the singular homology are obtained for generalized Dold spaces $P(S,X)=S\times X/\langle \theta\rangle$, where $\theta=\alpha\times \sigma$, $\alpha:S\to S$ is a fixed point free involution and $\sigma:X\to X$ is an involution with $\mathrm{Fix}(\sigma)\ne \emptyset,$ for a much wider class of spaces $S$ and $X$.

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arXiv2021

Cohomology of generalized Dold spaces

Let $(X,J) $ be an almost complex manifold with a (smooth) involution $σ:X\to X$ such that fix($σ$) is non-empty. Assume that $σ$ is a complex conjugation, i.e, the differential of $σ$ anti-commutes with $J$. The space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim (-v,σ(x))$ was referred to as a generalized Dold manifold. The above definition admits an obvious generalization to a much wider class of spaces where $X, S$ are arbitrary topological spaces. The resulting space $P(S,X)$ will be called a generalized Dold space. When $S$ and $X$ are CW complexes satisfying certain natural requirements, we obtain a CW-structure on $P(S,X)$. Under certain further hypotheses, we determine the mod $2$ cohomology groups of $P(S,X)$. We determine the $\mathbb Z_2$-cohomology algebra when $X$ is (i) a torus manifold whose torus orbit space is a homology polytope, (ii) a complex flag manifold. One of the main tools is the Stiefel-Whitney class formula for vector bundles over $P(S,X)$ associated to $σ$-conjugate complex bundles over $X$ when the $S$ is a paracompact Hausdorff topological space, extending the validity of the formula, obtained earlier by Nath and Sankaran, in the case of generalized Dold manifolds.

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