Cohomology of colorings of cycles

模 $p$ 上同调运算的几何构造

Geometric Constructions of Mod $p$ Cohomology Operations

本文基于Brown可表示性和Almgren同构定理，构造了从球面循环空间到圆盘相对循环空间的显式映射，用以表示模$p$上同调中的所有Steenrod平方、Steenrod幂和Bockstein同态。

Geometric cohomology frames on Hausmann-Holm-Puppe conjugation spaces

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…

Algebraic Cycles Representing Cohomology Operations

In this paper we show that certain universal homology classes which are fundamental in topology are algebraic. To be specific, the products of Eilenberg-MacLane spaces ${\cal K}_{2q} \equiv K({\Bbb Z},2) \times K({\Bbb Z}, 4) \times ... \times K({\Bbb Z}, 2q)$ have models which are limits of complex projective varieties. Precisely, we have ${\cal K}_{2q} = \lim_{d\to\infty}{\cal C}_d^q({\Bbb P}^n)$ where ${\cal C}_d^q({\Bbb P}^n)$ denotes the Chow variety of effective cycles of codimension $q$ and degree $d$ on ${\Bbb P}^n$. It is natural to ask which elements in the homology of ${\cal K}_{2q}…

On the cohomology rings of real flag manifolds: Schubert cycles

We give an algorithm to compute the integer cohomology groups of any real partial flag manifold, by computing the incidence coefficients of the Schubert cells. For even flag manifolds we determine the integer cohomology groups, by proving that any torsion class has order 2 (generalizing a result of Ehresmann). We conjecture this to hold for any real flag manifold. We obtain results concerning which Schubert varieties represent integer cohomology classes, their structure constants and how to express them in terms of characteristic classes. For even flag manifolds and Grassmannians we also descr…

The vanishing cohomology of non-isolated hypersurface singularities

We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower bound in dimension. Guided by geometric results, we alternately use the nearby and vanishing cycle functors to derive information about the Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead to the description of the reduced cohomology groups, except the top two, in terms of the vanishing cohomology of the nearby section. We use it …

The real cycle class map

The classical cycle class map for a smooth complex variety sends cycles in the Chow ring to cycles in the singular cohomology ring. We study two cycle class maps for smooth real varieties: the map from the I-cohomology ring to singular cohomology induced by the signature, and a new cycle class map defined on the Chow-Witt ring. For both maps, we establish basic compatibility results like compatibility with pullbacks, pushforwards and cup products. As a first application of these general results, we show that both cycle class maps are isomorphisms for cellular varieties.