We show that under standard assumptions on the isotropy groups of an integer GKM manifold, the equivariant Stiefel-Whitney classes of the action are determined by the GKM graph. This is achieved via a GKM-style description of the equivariant cohomology with coefficients in a finite field $\mathbb Z_{p}$ even though in this setting the restriction map to the fixed point set is not necessarily injective. This closes a gap in our argument why the GKM graph of a $6$-dimensional integer GKM manifold determines its nonequivariant diffeomorphism type. We introduce combinatorial Stiefel-Whitney classes of GKM graphs and use them to derive a nontrivial obstruction to realizability of GKM graphs in dimension $8$ and higher.
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 $2^q$.
We determine the Stiefel-Whitney classes of the second exterior representation and the spin representation of Spin(15), which are useful to calculate the mod 2 cohomology of the classifying space of the exceptional Lie group E_8.
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 determine a nice simple formula for the largest Euclidean space for which there is an orientable n-manifold with a nonimmersion detected by Stiefel-Whitney classes. For Spin manifolds, we prove the analogue of the upper bound and establish the complete answer for n<24 and n=33,34. Results similar to many of these were obtained some 50 years ago, but in a much less tractable form. The sharp results for Spin manifolds require detailed calculations of ko-homology groups of mod-2 Eilenberg-MacLane spaces.
The cohomology ring of a finite group, with coefficients in a finite field, can be computed by a machine, as Carlson has showed. Here "compute" means to find a presentation in terms of generators and relations, and involves only the underlying (graded) ring. We propose a method to determine some of the extra structure: namely, Stiefel-Whitney classes and Steenrod operations. The calculations are explicitly carried out for about one hundred groups (the results can be consulted on the Internet). Next, we give an application: thanks to the new information gathered, we can in many cases determine …
We prove that the Stiefel-Whitney classes of a moment-angle manifold, not necessarily smooth, are trivial. We also consider Stiefel-Whitney classes of the partial quotient of a moment-angle manifold.
This paper determines which Stiefel-Whitney numbers can be defined for singular varieties compatibly with small resolutions. First an upper bound is found by identifying the F_2-vector space of Stiefel-Whitney numbers invariant under classical flops, equivalently by computing the quotient of the unoriented bordism ring by the total spaces of RP^3 bundles. These Stiefel-Whitney numbers are then defined for any real projective normal Gorenstein variety and shown to be compatible with small resolutions whenever they exist. In light of Totaro's result [Tot00] equating the complex elliptic genus wi…
We describe the Stiefel-Whitney classes (SWCs) of orthogonal representations $π$ of the finite special linear groups $G=\text{SL}(2,\mathbb F_q)$, in terms of character values of $π$. From this calculation, we can answer interesting questions about SWCs of $π$. For instance, we determine the subalgebra of $H^*(G,\mathbb Z/2\mathbb Z)$ generated by the SWCs of orthogonal $π$, and we also determine which $π$ have nontrivial mod $2$ Euler class.
We propose a definition of persistent Stiefel-Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual Čech filtration of such a subset can be endowed with a vector bundle structure, that we call a Čech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its persistent Stiefel-Whitney classes. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial appro…
Let $(X,c_X)$ be a convex projective surface equipped with a real structure. The space of stable maps $\bar{\mathcal{M}}_{0,k}(X,d)$ carries different real structures induced by $c_X$ and any order two element $τ$ of permutation group $S_k$ acting on marked points. Each corresponding real part $\R_τ\bar{\mathcal{M}}_{0,k}(X,d)$ is a real normal projective variety. As the singular locus is of codimension bigger than two, these spaces thus carry a first Stiefel-Whitney class for which we determine a representative in the case $k=c_1(X)d-1$ where $c_1(X)$ is the first Chern class of $X$. Namely, …
We consider quotients of complex Stiefel manifolds by finite cyclic groups whose action is induced by the scalar multiplication on the corresponding complex vector space. We obtain a description of their tangent bundles, compute their mod p cohomology and obtain estimates for their span (with respect to their standard differentiable structure). We compute the Pontrjagin and Stiefel-Whitney classes of these manifolds and give applications to their stable parallelizability.
Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways these can be used to obtain lower bounds for the chromatic numbers of graphs, presented in a recent series of papers \cite{BK03a,BK03b,BK03c,CK1,CK2,K4,K5}. The structural theory is developed and put in the historical context, culminating in the proof of the Lovász Conjecture, which can be stated as follows: For a graph G, such that the complex Hom(C_{2r+1}…
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 $V_k(\mathbb F^n)$, $\mathbb F=\mathbb R,\mathbb C,\mathbb H$.
Using a combinatorial description of Stiefel-Whitney classes of closed flat manifolds with diagonal holonomy representation, we show that no Hantzsche-Wendt manifold of dimension greater than three does not admit a spin$^c$ structure.
For each integer $d$ at least two, we construct non-spin closed oriented flat manifolds with holonomy group $\mathbb Z_2^d$ and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel-Whitney classes.
Let h be a Real bundle, in the sense of Atiyah, over a space X. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that BU is equipped with a structure of conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern classes in the Z/2-equivariant cohomology of X with twisted integer coefficients. We show that these classes determine the (non-equivariant) Chern classes of h, forgetting the involution on X, and the Stiefel-Whitney classes of the real bundle of fixed points.
Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases
Department of Applied Mathematics I, University of Sevilla（塞维利亚大学应用数学系）；Graduate School of Mathematical Sciences, The University of Tokyo（东京大学数学科学研究生院）
