Related papers: On trivialities of Chern classes

A space $X$ is called $W$-trivial if for every vector bundle $\xi$ over $X$, the total Stiefel-Whitney class $W(\xi)= 1$. In this article we shall investigate whether the suspensions of Dold manifolds, $\s^k D(m,n)$, is $W$-trivial or not.

A space $X$ is said to be $C$-trivial if the total Chern class $c(\alpha)$ equals $1$ for every complex vector bundle $\alpha$ over $X$. In this note we give a complete homological classification of $C$-trivial closed smooth manifolds of…

A space $X$ is $W$-trivial if for every real vector bundle $\alpha$ over $X$ the total Stiefel-Whitney class $w(\alpha)$ is 1. It follows from a result of Milnor that if $X$ is an orientable closed smooth manifold of dimension $1,2,4$ or…

Let ${\mathbb F}_0$ be an algebraically closed field, with $char({\mathbb F}_0)=0$. In this article, for prime numbers $p\geq 2$, we construct smooth affine algebras $B$ over ${\mathbb F}_0$, with $\dim B=p+2$. Further, we construct…

We prove that Chern classes in continuous $\ell$-adic cohomology of automorphic bundles associated to representations of $G$ on a projective Shimura variety with data $(G,X)$ are trivial rationally. It is a consequence of Beilinson's…

This short note summarizes a number of facts about the ring $K^0(X)$ for $X$ a $4$-dimensional CW-complex. Unusual features of this dimension are that every complex vector bundle is determined up to stable isomorphism by its Chern classes,…

Let $M_X(r,\xi)$ be the moduli space of stable vector bundles, on a smooth complex projective curve $X$, of rank $r$ and fixed determinant $\xi$ such that $\deg(\xi)$ is coprime to $r$. If $E$ is a vector bundle $M_X(r,\xi)$ whose…

Using the theory of resolving classes, we show that if $X$ is a CW complex of finite type such that $\map_*(X, S^{2n+1})\sim *$ for all sufficiently large $n$, then $\map_*(X, K) \sim *$ for every simply-connected finite-dimensional CW…

In this paper, we give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos. We also prove a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert…

Let $X^k_{m,n}=\Sigma^k (\mathbb R\mathbb P^m/\mathbb R\mathbb P^n)$. In this note we completely determine the values of $k,m,n$ for which the total Stiefel-Whitney class $w(\xi)=1$ for any vector bundle $\xi$ over $X^k_{m,n}$.

Let $X$ be a finite dimensional compact metrizable space. We study a technique which employs semiprojectivity as a tool to produce approximations of $C(X)$-algebras by $C(X)$-subalgebras with controlled complexity. The following…

Let $(X,0)$ be an isolated complete intersection complex singularity ($X$ can also be smooth at 0). Let $K$ be its link, $\cal X$ its canonical contact structure and $\D_X$ the complex vector bundle associated to $\cal X$. We prove that the…

In the previous part of this diptych, we defined the notion of an admissible simplicial connection, as well as explaining how H.I. Green constructed a resolution of coherent analytic sheaves by locally free sheaves on the \v{C}ech nerve.…

In this paper, complex vector bundles of rank $r$ over $8$-dimensional spin$^{c}$ manifolds are classified in terms of the Chern classes of the complex vector bundles and the cohomology ring of the manifolds, where $r = 3$ or $4$. As an…

Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation…

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show,…

Let $k$ be an algebraically closed base field of characteristic $0$ and let $\alpha_{1}, \alpha_{2}, \alpha_{3}, d \geq 2$ be integers such that $\alpha_{1}, \alpha_{2}, \alpha_{3}$ are pairwise coprime and $gcd (\alpha_{1},d-1) = 1$. Then…

Let $Y$ be a normal and projective variety over an algebraically closed field $k$ and $V$ a vector bundle over $Y$. We prove that if there exist a $k$-scheme $X$ and a finite surjective morphism $g:X\to Y$ that trivializes $V$ then $V$ is…

We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only projective manifolds with trivial Chern classes. By Yau' s celebrated result, compact K\"ahler manifolds with trivial Chern classes…

We apply Weiss calculus to compute the number of topological complex vector bundles of rank $n-2$ with vanishing Chern classes over $\mathbb{C}P^n$ for $n>3$, as given by the list $1, 1, 12, 2, 1, 3, 2, 2, 3, 1, 4, 6, 1, 1, 6, 2, 1, 3, 4,…