19 pages. A minor error in Prop. 2.5(iii) had been corrected. There was a gap in the proof of Theorem 1.2 which has been corrected. Other minor typos were corrected
Let $X$ be a smooth manifold with a (smooth) involution $σ:X\to X$ such that $Fix(σ)\ne \emptyset$. We call the space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim (-v,σ(x))$ a generalized Dold manifold. When $X$ is an almost complex manifold and the differential $Tσ: TX\to TX$ is conjugate complex linear on each fibre, we obtain a formula for the Stiefel-Whitney polynomial of $P(m,X)$ when $H^1(X;\mathbb{Z}_2)=0$. We obtain results on stable parallelizability of $P(m,X)$ and a very general criterion for the (non) vanishing of the unoriented cobordism class $[P(m,X)]$ in terms of the corresponding properties for $X$. These results are applied to the case when $X$ is a complex flag manifold.
It is well known that Dold and Milnor manifolds give generators for the unoriented bordism algebra ${\frak{N}}_*$ over ${\Bbb{Z}}_2$. The purpose of this paper is to determine those Milnor manifolds which represent the same bordism classes in ${\frak{N}}_*$ as their Dold counterparts.
D. Davis introduced projective product spaces in 2010 as a generalization of real projective spaces and discussed some of their topological properties. On the other hand, Dold manifolds were introduced by A. Dold in 1956 to study the generators of the non-oriented cobordism ring. Recently, in 2019, A. Nath and P. Sankaran made a modest generalization of Dold manifolds. In this paper we simultaneously generalize both the notions of projective product spaces and Dold manifolds, leading to infinitely many different classes of new smooth manifolds. Our main goal will be to study the integral homol…
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.
The Dold manifold $ P(m,n)$ is the quotient of $S^m \times \mathbb{C}P^n$ by the free involution that acts antipodally on $ S^m $ and by complex conjugation on $ \mathbb{C}P^n $. In this paper, we investigate free actions of finite groups on products of Dold manifolds. We show that if a finite group $ G $ acts freely and mod 2 cohomologically trivially on a finite-dimensional CW-complex homotopy equivalent to ${\displaystyle \prod_{i=1}^{k} P(2m_i,n_i)}$, then $G\cong (\mathbb{Z}_2)^l$ for some $l\leq k$. This is achieved by first proving a similar assertion for $ \displaystyle \prod_{i=1}^{k}…
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 req…
Let $(X,J) $ be an almost complex manifold with a (smooth) involution $σ:X\to X$ such that $Fix(σ)\neq \emptyset$. 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 $(v,x)\sim (-v,σ(x))$ is known as a generalized Dold manifold. Suppose that a group $G\cong \mathbb Z_2^s$ acts smoothly on $X$ such that $g\circ σ=σ\circ g$ for all $g\in G$. Using the action of the diagonal subgroup $D=O(1)^{m+1}\subset O(m+1)$ on the sphere $\mathbb S^{m}$ for which there are only finitely many pairs of antipodal p…
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 nonexisten…
When the cohomology ring of a generalized Bott manifold with $\mathbb{Q}$-coefficient is isomorphic to that of a product of complex projective spaces $\mathbb{C}P^{n_i}$, the generalized Bott manifold is said to be $\mathbb{Q}$-trivial. We find a necessary and sufficient condition for a generalized Bott manifold to be $\mathbb{Q}$-trivial. In particular, every $\mathbb{Q}$-trivial generalized Bott manifold is diffeomorphic to a $\prod_{n_i>1}\mathbb{C}P^{n_i}$-bundle over a $\mathbb{Q}$-trivial Bott manifold.
We show that all generalized (real) Bott manifolds which are (small covers) quasitoric manifolds over a product of simplices $Δ^{n_1}\times\cdots\timesΔ^{n_r}\timesΔ^{1}$ are always boundaries of some manifolds. But these manifolds with the natural $(\mathbb{Z}_2)^n$ action do not necessarily bound equvariantly. In addition, we can construct some examples of null-cobordant but not orientedly null-cobordant manifolds among quasitoric manifolds.
The question of paralleizability and stable parallelizability of a family of manifolds obtained as a quotients of circle action on the complex Stiefel manifolds are studied and settled.
To a direct sum of holomorphic line bundles, we can associate two fibrations, whose fibers are, respectively, the corresponding full flag manifold and the corresponding projective space. Iterating these procedures gives, respectively, a flag Bott tower and a generalized Bott tower. It is known that a generalized Bott tower is a toric manifold. However a flag Bott tower is not toric in general but we show that it is a GKM manifold, and we also show that for a given generalized Bott tower we can find the associated flag Bott tower so that the closure of a generic torus orbit in the latter is a b…
In this paper, we give a necessary and sufficient condition for a generalized real Bott manifold to have a Spin structure in terms of column vectors of the associated matrix. We also give an interpretation of this result to the associated acyclic $ω$-weighted digraphs. Using this, we obtain a family of real Bott manifolds that does not admit Spin Structure.
Generalized Bott manifolds (over $\mathbb C$ and $\mathbb R$) have been defined by Choi, Masuda and Suh. In this article we extend the results of arXiv:1609.05630 on the topology of real Bott manifolds to generalized real Bott manifolds. We give a presentation of the fundamental group, prove that it is solvable and give a characterization for it to be abelian. We further prove that these manifolds are aspherical only in the case of real Bott manifolds and compute the higher homotopy groups. Furthermore, using the presentation of the cohomology ring with $\mathbb Z_2$-coefficients, we derive a …
We investigate when two generalized real Bott manifolds of height 2 have isomorphic cohomology rings with Z/2 coefficients and also when they are diffeomorphic. It turns out that cohomology rings with Z/2 coefficients do not distinguish those manifolds up to diffeomorphism in general. This gives a counterexample to the cohomological rigidity problem for real toric manifolds posed in \cite{ka-ma08}. We also prove that generalized real Bott manifolds of height 2 are diffeomorphic if they are homotopy equivalent.
In this paper we develop a theory of volume polynomials of generalized virtual polytopes based on the study of topology of affine subspace arrangements in a real Euclidean space. We apply this theory to obtain a topological version of the BKK Theorem, the Stanley-Reisner and Pukhlikov-Khovanskii type descriptions for cohomology rings of generalized quasitoric manifolds.
We show that the classical example $X$ of a 3-dimensional generalized manifold constructed by van Kampen is another example of not homologically locally connected (i.e. not HLC) space. This space $X$ is not locally homeomorphic to any of the compact metrizable 3-dimensional manifolds constructed in our earlier paper which are not HLC spaces either.
We give a direct proof of a cancellation formula raised in [7] on the level of differential forms. We also obtain more cancellation formulas for even dimensional Riemannian manifolds with a complex line bundle involved. Relations among these cancellation formulas are discussed.
We demonstrate how by using the intersection theory to calculate the cohomology of $G_2$-manifolds constructed by using the generalized Kummer construction. For one example we find the generators of the rational cohomology ring and describe the product structure.
We define Aeppli and Bott-Chern cohomology for bi-generalized complex manifolds and show that they are finite dimensional for compact bi-generalized Hermitian manifolds. For totally bounded double complexes $(A, d', d'')$, we show that the validity of $d'd''$-lemma is equivalent to having the same dimension of several cohomology groups. Some calculations of Bott-Chern cohomology groups of some bi-generalized Hermitian manifolds are given.
