Samik Basu, Debanil Dasgupta
In this paper, we compute the BP-cohomology of complex projective Stiefel manifolds. The method involves the homotopy fixed point spectral sequence, and works for complex oriented cohomology theories. We also use these calculations and BP-operations to prove new results about equivariant maps between Stiefel manifolds.
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In this paper, we compute the $RO(\mathbb{Z}/2)$-graded equivariant cohomology of rotation groups and Stiefel manifolds with particular involutions.
The main objective of this paper is to analyze the $p$-local homotopy type of the complex projective Stiefel manifolds, and other analogous quotients of Stiefel manifolds. We take the cue from a result of Yamaguchi about the $p$-regularity of the complex Stiefel manifolds which lays down some hypotheses under which the Stiefel manifold is $p$-locally a product of odd dimensional spheres. We show that in many cases, the projective Stiefel manifolds are $p$-locally a product of a complex projective space and some odd dimensional spheres. As an application, we prove that in these cases, the $p$-r…
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.
We compute the cohomology of the right generalised projective Stiefel manifolds and use it to find bounds on the rank of the complementary bundle for certain vector bundles. Further the cohomology computations are also used to find bounds on the span and on the immersibility of the manifolds in certain cases.
Some Observations on projective Stiefel Manifolds
本文计算了射影Stiefel流形和翻转Stiefel流形的上特征秩，给出了cup长度的界，并利用Fadell-Husseini指标提供了四元数Stiefel流形间$S^3$-映射存在的必要条件。
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 …
Mod 2 cohomology of 2-configuration space of a closed surface and Stiefel--Whitney class
本文计算闭曲面的有序与无序2-配置空间的模2上同调，并由此确定Stiefel--Whitney高度在可定向与不可定向情形下分别为2或3。
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.
Cohomology of spaces of Hopf equivariant maps of spheres
本文证明了球面Hopf等变映射空间的有理上同调环与相应Stiefel流形自然同构，并指出其整上同调群的自然映射是满射但非单射。
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 $V_k(\mathbb F^n)$, $\mathbb F=\mathbb R,\mathbb C,\mathbb H$.
A Study of topology of the Flip Stiefel Manifolds
本文引入了由阶为 2 循环群对实 Stiefel 流形坐标成对翻转作用产生的新商流形，计算了其切丛、模 2 上同调及 Stiefel Whitney 类，并应用于稳定生成、平行化、等变映射与拓扑组合学。
Milnor-Witt motivic decomposition of Stiefel varieties
本文计算了Stiefel流形的整MW-动机上同调群，并建立了其整MW-动机分解，从而证明了先前工作中的猜想。
On the Stiefel-Whitney classes of GKM manifolds
本文证明在标准假设下，整数GKM流形的等变Stiefel-Whitney类由GKM图决定，并引入组合Stiefel-Whitney类，为高维GKM图可实现性提供非平凡障碍。
Heights of Stiefel--Whitney classes and zero-divisor cup-length of some Grassmann manifolds
本文计算了某些Grassmann流形上Stiefel-Whitney类的高度，并利用模2上同调计算得出拓扑复杂度的下界，扩展了Rusin关于Grassmann流形模2上同调长度的研究。
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, …
The integral cohomology ring of the complement of an arrangement of linear subspaces of a finite dimensional complex projective space is determined by combinatorial data, i.e. the intersection poset and the dimension function.
In this paper we prove, that none of the three combinatorial 8- manifolds on 15 vertices constructed by Brehm and Kuhnel, each of which is a cohomology quaternionic projective plane, can be combinatorially embedded in the Euclidean 12-space, though they have tight polyhedral embeddings in Euclidean 14-space. This extends a similar method already known for the nonembeddings of real and complex projective planes.
The cohomology of projective unitary groups
本文通过显式构造生成元计算射影酉群PU(n)的整数上同调环，揭示其结构特性。
We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations of $C_2$, using ordinary cohomology graded on representations of the fundamental groupoid, with coefficients in the Burnside ring Mackey functor. This extension of the $RO(C_2)$-graded theory allows for the definition of Euler classes, which are used as generators of the cohomology of the projective spaces. As an application, we give an equivariant version of Bezout's theorem.
We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with simple and twisted coefficients---of the dihedral group of order 8 (in the case of unordered configurations) and the elementary abelian 2-group of rank 2 (in the case of ordered configurations). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimension 2^i + d for d=0,1,2.
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…
Kitchloo and Wilson have used the homotopy fixed points spectrum ER(2) of the classical complex-oriented Johnson-Wilson spectrum E(2) to deduce certain non-immmersion results for real projective spaces. ER(n) is a $2^{n+2}(2^n-1)$-periodic spectrum. The key result to use is the existence of a stable cofibration $Σ^{λ(n)}ER(n) \rightarrow ER(n) \rightarrow E(n)$ connecting the real Johnson-Wilson spectrum with the classical one. The value of $λ(n)$ is $2^{2n+1}-2^{n+2}+1$. We extend Kitchloo-Wilson's results on non-immersions of real projective spaces by computing the second real Johnson-Wilson…
Let X be a space and write LX for its free loop space equipped with the action of the circle group T given by dilation. We compute the equivariant cohomology H^*(LX_hT; Z/p) as a module over H^*(BT; Z/p) when X=CP^r for any positive integer r and any prime number p. The computation implies that the associated mod p Serre spectral sequence collapses from the E_3-page.
Let R be a commutative ring containing 1/2. We compute the R-cohomology ring of the configuration space F(m,k) of k ordered points in the m-dimensional real projective space. The method uses the observation that the orbit configuration space of k ordered points in the m-dimensional sphere (with respect to the antipodal action) is a 2^k-fold covering of F(m,k). This implies that, for odd m, the Leray spectral sequence for the inclusion of F(m,k) in the k-fold Cartesian self power of the m-dimensional real projective space collapses after its first non-trivial differential, just as it does when …
Cohomology algebra of orbit spaces of free involutions on the product of projective space and 4-sphere
研究具有射影空间与4-球面乘积模2上同调的有限型空间在自由对合下的商空间上同调代数，并推导等变映射存在性结论。
We use Brown-Peterson cohomology to obtain lower bounds for the higher topological complexity, TC_k(RP^n), of real projective spaces, which are often much stronger than those implied by ordinary mod-2 cohomology.
In this paper, we first discuss the topological properties of projective Stiefel manifolds, we compute their cohomology rings and classify their cohomology endomorphisms; Then by embedding the flag manifold of a classical Lie group into its corresponding infinite dimensional projective Stiefel manifold(which is homotopic to the product of infinite dimensional complex projective space $\mathbb{C}P^{\infty}$), we define the Schubert polynomials and dual Schubert polynomials. Finally we discuss the property and the computation of these polynomials.
We compute the integral cohomology ring of configuration spaces of two points on a given real projective space. Apart from an integral class, the resulting ring is a quotient of the known integral cohomology of the dihedral group of order 8 (in the case of unordered configurations, thus has only 2- and 4-torsion) or of the elementary abelian 2-group of rank 2 (in the case of ordered configurations, thus has only 2-torsion). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimensions of the form 2^i+j for non-negative i and j wi…
Equivariant Cohomology of Projective Spaces
本文计算了射影空间在整数系数下的等变同调与上同调，证明循环群情形下其胞腔滤过给出谱的楔和分裂，并推广到四元数情形、复共轭作用及素数幂阶群，进而描述 $S^1$ 和 $S^3$ 分类空间的等变同调与上同调。
This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S3 and the orbit space X/G is the integral or mod 2 cohomology quaternion projective space HPn. We have proved that X is the integral or mod 2 cohomology (4n+3)-sphere or the product of 3-sphere and quaternion projective space HPn. Similar results for G = S1 actions are also discussed.
The Real-oriented cohomology of infinite stunted projective spaces
本文计算无限截断射影谱的ER-上同调，揭示其与实谱的余纤维关系，并用于计算Mahowald不变量。
Let $Λ(\mathbb{C}P^n)$ denote the free loop space of the complex projective space $\mathbb{C}P^n$, i. e. $\mathbb{C}P^n$ is the projective space of the vector space $\mathbb{C}^{n+1}$ of dimension $n+1$ over the complex numbers $\mathbb{C}$ and $Λ(\mathbb{C}P^n)$ is the function space $\mathrm{map}(S^1,\mathbb{C}P^n)$ of unbased maps from a circle $S^1$ into $\mathbb{C}P^n$ topologized with the compact open topology. In this note we show that despite the fact that the natural fibration $Ω(\mathbb{C}P^n)\hookrightarrow Λ(\mathbb{C}P^n)\stackrel{eval}{\longrightarrow}\mathbb{C}P^n$ has a cross s…
For the model two-complex $K$ of the group presentation $\mathcal{P}=\langle x,y\,|\,x^{k+1}yxy \rangle$, with $k\geq1$ odd, we describe representatives for all free and based homotopy classes of maps from $K$ into the real projective plane and we classify the homotopy classes containing only surjective maps. With this approach we get an answer, for maps into the real projective plane, for a classical question in topological root theory, which is known so far, in dimension two, only for maps into the sphere, the torus and the Klein bottle. The answer follows by proving that for all $k\geq1$ od…
A torus manifold is a closed smooth manifold of dimension $2n$ having an effective smooth $T^n = (S^1)^n$-action with non-empty fixed points. Petrie \cite{petrie:1973} has shown that any homotopy equivalence between a complex projective space $\CP^n$ and a torus manifold homotopy equivalent to $\CP^n$ preserves their Pontrjagin classes. A \emph{generalized Bott manifold} is a closed smooth manifold obtained as the total space of an iterated complex projective space bundles over a point, where each fibration is a projectivization of the Whitney sum of a finite many complex line bundles. For ins…
Chen and Ruan [6] defined a very interesting cohomology theory for orbifolds, which is now called Chen-Ruan cohomology. The primary objective of this paper is to compute the Chen-Ruan cohomology rings of the weighted projective spaces, a class of important spaces in physics. The classical tools (Chen-Ruan cohomology, toric varieties, the localization technique) which have been proved to be successful are used to study the orbifold cohomology of weighted projective spaces. Given a weighted projective space ${\bf P}^{n}_{q_{0}, >..., q_{n}}$, we determine all of its twisted sectors and the corre…
We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. We deduce that the ring is a perfect invariant, and prove a Chern class formula for weighted projective bundles.
Finitistic Spaces with Orbit Space a Product of Projective Spaces
该文研究Z₂自由作用下有限型空间与其轨道空间模2上同调的逆问题，证明了轨道空间为两类射影空间乘积时原空间的上同调结构，并给出射影空间乘积上自由对合的部分逆结论。
Let $X$ be a finitistic space having the mod 2 cohomology algebra of the product of two projective spaces. We study free involutions on $X$ and determine the possible mod 2 cohomology algebra of orbit space of any free involution, using the Leray spectral sequence associated to the Borel fibration $X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}$. We also give an application of our result to show that if $X$ has the mod 2 cohomology algebra of the product of two real projective spaces (respectively complex projective spaces), then there does not exist any $\mathbb{Z}_2$-equ…
For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets). Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of "conjugation spaces", where the ring homomorphisms arise naturally from the existence of what they call "cohomology…
We compute the Betti numbers and describe the cohomology algebras of the ordered and unordered configuration spaces of three points in complex projective spaces, including the infinite dimensional case. We also compute these invariants for the configuration spaces of three collinear and non-collinear points.
We study the mod p cohomology of the classifying space of the projective unitary group PU(p). We first proof that old conjectures due to J.F. Adams, and Kono and Yagita about the structure of the mod p cohomology of classifying space of connected compact Lie groups held in the case of PU(p). Finally, we proof that the classifying space of the projective unitary group PU(p) is determined by its mod p cohomology as an unstable algebra over the Steenrod algebra for p>3, completing previous works for the cases p=2,3.
Let $\mathbf{B}PU_{n}$ be the classifying space of $PU_n$, the projective unitary group of order $n$, for $n>1$. We use the Serre spectral sequence associated to a fiber sequence $\mathbf{B}U_n\rightarrow\mathbf{B}PU_n\rightarrow K(\mathbb{Z},3)$ to determine the ring structure of $H^{*}(\mathbf{B}PU_{n}; \mathbb{Z})$ up to degree $10$, as well as a family of distinguished elements of $H^{2p+2}(\mathbf{B}PU_{n}; \mathbb{Z})$, for each prime divisor $p$ of $n$. We also study the primitive elements of $H^*(\mathbf{B}U_n;\mathbb{Z})$ as a comodule over $H^*(K(\mathbb{Z},2);\mathbb{Z})$, where the…
Persistent Stiefel-Whitney Classes of Tangent Bundles
研究如何从有限点样本计算切丛的施蒂费尔 - 惠特尼类，核心方法是构建单纯复形滤链计算持久上同调并应用吴公式，该算法时间复杂度为多项式，能在多种情况下准确计算出切丛的此类拓扑不变量。
Čech cohomology of infinite projective spaces, flag manifolds, and related spaces
本文通过计算无限射影空间和旗 manifold 的 Čech 上同调环，揭示了可数无限产品空间的上同调结构。
We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundle…
In this paper, for any Milnor hypersurface we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalised Buchstaber-Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.
Cohomology of hypersurfaces of weighted projective space and the intersection form on $H^2$
本文通过计算加权射影空间超曲面的第二上同调相交形式并描述低次整系数上同调群，为识别奇异 Fano 簇光滑化得到的 Fano 流形提供了显式工具。
This paper is concerned with the primitive cohomology of a smooth projective hypersurface considered as a linear representation for its automorphism group. Using the Lefschetz-Riemann-Roch formula, the character of this representation is described on each piece of the Hodge decomposition. A consequence about the existence of smooth symmetric hypersurfaces that are stable under the standard irreducible permutation representation of the symmetric group on homogeneous coordinates is drawn.
Motivic cohomology of the Nisnevich classifying space of even Clifford groups
本文计算了分裂偶 Clifford 群的 Nisnevich 分类空间的模 2 动机上同调环，并指出其与 spin 群情形的主要差异在于第二个精细 Stiefel-Whitney 类非平凡。
Mod 2 representation of the symmetric group of order 2 over cohomology groups of 2-configuration space of torus
本文计算了 $d$-环面有序2-构型空间奇异上同调群上2阶对称群的模2表示，并据此确定了其 Stiefel-Whitney 高度及 $d=2,3$ 时无序构型空间的上同调群模结构。
Borel-Hirzebruch type formula for the graph equivariant cohomology of a projective bundle over a GKM-graph
本文定义GKM理论中的腿丛及其射影化，证明射影化图等变上同调的Borel-Hirzebruch型公式，并说明其可实现Guillemin-Sabatini-Zara意义下的射影GKM纤维丛。
We compute the integral torus-equivariant cohomology ring for weighted projective space for two different torus actions by embedding the cohomology in a sum of polynomial rings $\oplus_{i=0}^n \Z[t_1, t_2,..., t_n]$. One torus action gives a result complementing that of Bahri, Franz, and Ray. For the other torus action, each basis class for weighted projective space is a multiple of the basis class for ordinary projective space; we identify each multiple explicitly. We also give a simple formula for the structure constants of the equivariant cohomology ring of ordinary projective space in term…
We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the LS category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology rin…
The mapping class group $Γ^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of $\mathbb R {\rm P}^2$, we analize the Serre spectral sequence of a fiber bundle $F_k(\mathbb R {\rm P}^2)/Σ_k \to X_k \to BSO(3)$ where $X_k$ is a $K(Γ^k(\mathbb R {\rm P}^2),1)$ and $F_k(\mathbb R {\rm P}^2)/Σ_k$ denotes the configuration space of unordered $k$-tuples of distinct points in $\mathbb R {\rm P}^2$. As a consequence, we express the mod-2 cohomol…
Let X be the quasi-projective symplectic surface that is given by the total space of the invertible sheaf O(-2) over the projective line. Let Hilb X be the family of Hilbert schemes of points on X. We give and prove a closed formula expressing any multiplicative characteristic class evaluated on the Hilb X in terms of the standard Fock space description of the cohomology of the Hilb X. As a side result, we also deduce a formula for the Chern character of the tangent bundles of the Hilb X. The results found here are another step towards a complete description of the tangent bundle of the Hilber…
The present paper describes a relation between the quotient of the fundamental group of a smooth quasi-projective variety by its second commutator and the existence of maps to orbifold curves. It extends previously studied cases when the target was a smooth curve. In the case when the quasi-projective variety is a complement to a plane algebraic curve this provides new relations between the fundamental group, the equation of the curve, and the existence of polynomial solutions to certain equations generalizing Pell's equation. These relations are formulated in terms of the depth which is an in…
Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis neither on the schemes nor on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher $K$-theory and motivic cohomology as well as an analogue result for the relative cohomology of a morphism. These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriente…
We compute the rational cohomology of the moduli space of non-singular complex projective curves of genus 3 with two marked points.
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