Last edited by Bazuru
Thursday, July 16, 2020 | History

2 edition of Equivariant Stiefel-Whitney classes found in the catalog.

Equivariant Stiefel-Whitney classes

Robert C. Johnson

# Equivariant Stiefel-Whitney classes

## by Robert C. Johnson

Published .
Written in English

Subjects:
• Homology theory.

• Edition Notes

The Physical Object ID Numbers Statement by Robert Carl Johnson. Pagination 59 leaves, bound ; Number of Pages 59 Open Library OL17944877M

Schubert Varieties, Equivariant Cohomology and Characteristic Classes: IMPANGA 15 Jaroslaw Buczynski, Mateusz Michalek, Elisa Postinghel MPANGA stands for the activities of Algebraic Geometers at the Institute of Mathematics, Polish Academy of Sciences, including one of the most important seminars in algebraic geometry in Poland. The narrative arc of the course begins with Serre's work from the early 's: Fibrations, spectral sequences, and the cohomology of Eilenberg-Maclane spaces. As a prologue, we read about the construction and basic properties of Steenrod operations. These lead to a construction of the Stiefel-Whitney classes (from Milnor and Stasheff).

Abstract. We consider Chern classes, Stiefel-Whitney classes, and the Euler class from an axiomatic point of view. The uniqueness of the classes follows from the splitting principle, and the existence is derived using the bundle of projective spaces associated Author: Dale Husemoller. Stiefel-Whitney classes Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance.

A Concise Course in Algebraic Topology / Edition 2. by J. P. May, Peter J. May This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. The construction of the Stiefel-Whitney classes Brand: University of Chicago Press. 2. Characteristic classes for vector bundles 3. Stiefel-Whitney classes of manifolds 4. Characteristic numbers of manifolds 5. Thom spaces and the Thom isomorphism theorem 6. The construction of the Stiefel-Whitney classes 7. Chern, Pontryagin, and Euler classes 8. A glimpse at the general theory Chapter File Size: 1MB.

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### Equivariant Stiefel-Whitney classes by Robert C. Johnson Download PDF EPUB FB2

This is obvious by functoriality of the equivariant Stiefel-Whitney classes by considering the embedding of the trivial group (1) into G. 0 Proposition Assume X connected. Then W,i(E,p) is invariant by connected base change; it is denoted by wi(p) and called the ith geometric Stiefel- Whitney class of p.

Related concepts. orientation, Spin structure, w4-structure. Chern class. Pontryagin class, Wu class, one-loop anomaly polynomial I8. orthogonal calculus. References. Named after Eduard Stiefel and Hassler Whitney. Textbook accounts include. John Milnor, James D.

Stasheff, Characteristic Classes, Annals of Mathematics Stud Princeton University Press (). We develop a theory of equivariant Stiefel-Whitney classes for equivariant orthogonal vector bundles over schemes, parallel to Grothendieck's theory o Cited by: 7.

The cohomology ring of a point is the ring Z in degree 0. By homotopy invariance, this is also the cohomology ring of any contractible space, such as Euclidean space Rn.

The first cohomology group of the 2-dimensional torus has a basis given by the classes of the two circles shown. For a positive integer n, the cohomology ring of the sphere Sn. The Stiefel–Whitney classes are denoted by, and for a real vector bundle over a topological space, the class lies in.

These classes were introduced by E. Stiefel and H. Whitney and have the following properties. 1) For two real vector bundles over a common base, in other words, where is the complete Stiefel–Whitney class. The cohomology of flag manifolds is treated in detail (without spectral sequences), including the Equivariant Stiefel-Whitney classes book between Stiefel-Whitney classes and Schubert calculus.

More recent developments are also covered, including topological complexity, face spaces, equivariant Morse theory, conjugation spaces, polygon spaces, amongst : The obstruction to having a spin structure is a certain element [k] of H 2 (M, Z 2). For a spin structure the class [k] is the second Stiefel–Whitney class w 2 (M) ∈ H 2 (M, Z 2) of M.

Hence, a spin structure exists if and only if the second Stiefel–Whitney class w 2 (M) ∈ H. In this note, the first two Stiefel-Whitney classes of unoriented, oriented, and complex grassmannians are determined in terms of the Stiefel-Whitney classes of their tautological is achieved via the method mentioned at the end of Mark Grant's answer.

For the unoriented grassmannian $\operatorname{Gr}(m, m+n) = O(m+n)/(O(m)\times O(n))$, we have. Chapter 9. Stiefel-Whitney classes Trivializations and structures on vector bundles The class w1 – Orientability The class ˙w2 – Spin structures Deﬁnition and properties of Stiefel-Whitney classes Real ﬂag manifolds Deﬁnitions and Morse theory Cohomology rings The Chern classes, which are analogous to Stiefel-Whitney classes but related to complex representations rather than real ones, and which can be computed mod 2 from the Stiefel-Whitney classes, are always supported by algebraic varieties; this gives a \lower bound".

On the other hand, classes coming from the Chow ring are killed by certain SteenrodFile Size: KB. The total Stiefel-Whitney class of the canonical line bundle 1 n overPnisgivenby wp 1 n q 1 a whereadenotesthegeneratorofH pPn;Z 2q(cf.

Thm. Proof. Wehaveanobviousbundlemap Ep 1 1 q / 1 1 Ep 1 n q 1 n P1 incl /Pn Therefore 0 (S5) ˘ w 1 p 1q incl pw 1p 1 n qq and this shows that w 1p 1 n STIEFEL-WHITNEY CLASSES I. AXIOMS AND File Size: KB. The Stiefel–Whitney classes (and in fact, all characteristic classes) of a vector bundle give you information about the presence of trivial subbundles.

To be more precise, given a (real) vector bundle $p:E\rightarrow B$, with fibre [mat. Certain Stiefel-Whitney classes of manifolds with smooth, effective toral actions are shown to be computable in terms of Poincare duals of fixed point sets of isotropy subgroups. The cohomology of flag manifolds is treated in detail (without spectral sequences), including the relationship between Stiefel-Whitney classes and Schubert calculus.

More recent developments are also covered, including topological complexity, face spaces, equivariant Morse theory, conjugation spaces, polygon spaces, amongst others. ON THE STIEFEL-WHITNEY CLASSES OF A MANIFOLD. II Whitney class of an 8-dimensional manifold which can be nonzero.

It is known that W3 is the first obstruction to the existence of an al-most complex structure on Ms, and if W3 = Q, then Wi is the second obstruction.

Therefore if we remove a single point from any compact,File Size: KB. This book is an outcome of a recent () summer program in geometric combinatorics at the IAS/Park City Mathematics editors have done an excellent job in bringing together many leaders of the field and encouraging them to write expository lecture notes on various topics that expertly showcase the multi-faceted world of this vast and rapidly growing field of mathematics.

In their book [18, Appendix H], Guillemin–Ginzburg–Karshon discussed the problem of calculating the ring HG ∗ of equivariant Hamiltonian bordism classes of all unitary Hamil-tonian G-manifolds with integral equivariant cohomology classes 1 2π [ω−Φ], where Gis a torus.

With respect to Cited by: 1. Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class.

Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli.

The computation of Stiefel-Whitney classes. This course-tested book provides a valuable reference for algebraic topologists and includes foundational material essential for graduate study.

It seems from my (relatively short list) that if the top Stiefel-Whitney class is nonzero, there must be another nonzero class.

Recall that, at least for oriented manifolds, the top Stiefel-Whitney class is the mod 2 reduction of the Euler class, which is just the Euler characteristic times the fundamental class.STIEFEL-WHITNEY CLASSES smooth analytic varieties (of various dimensions) and ˜(’−1(X)) is odd for all x2X.

The chain s i(K)istheith Stiefel chain of the triangulation homology class w i(X)istheith Stiefel-Whitney homology class of the variety y w 0(X) ˜(X) (mod 2), where: H 0(X;Z=2Z)!Z=2Zis the augmentation homomorphism. Also, if Xhas pure dimension d,thenw.Get this from a library!

Mod two homology and cohomology. [J C Hausmann] -- Cohomology and homology modulo 2 helps the reader grasp more readily the basics of a major tool in algebraic topology. Compared to a more general approach to (co)homology this refreshing approach has.