Terms Of Characteristic Classes Research Articles

Parametrized topological complexity of sphere bundles

Parametrized motion planning algorithms \cite{CFW} have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles. Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly compute the parametrized topological complexity in many examples and show that it may assume arbitrarily large values.

Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum \(E_n\), whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that \(E_n\) is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group \({\mathbb {G}}_n\) of the formal group in question. In this paper we find that the \({\mathbb {G}}_n\)-equivariant dual of \(E_n\) is in fact \(E_n\) twisted by a sphere with a non-trivial (when \(n>1\)) action by \({\mathbb {G}}_n\). This sphere is a dualizing module for the group \({\mathbb {G}}_n\), and we construct and study such an object \(I_{{\mathcal {G}}}\) for any compact p-adic analytic group \({\mathcal {G}}\). If we restrict the action of \({\mathcal {G}}\) on \(I_{{\mathcal {G}}}\) to certain type of small subgroups, we identify \(I_{{\mathcal {G}}}\) with a specific representation sphere coming from the Lie algebra of \({\mathcal {G}}\). This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of \(E_n^{hH}\) for select choices of p and n and finite subgroups H of \({\mathbb {G}}_n\).

Principal bundles on two-dimensional CW-complexes with disconnected structure group

AbstractGiven any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that $\pi_0G$ is abelian.

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 that this holds 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 describe Schubert calculus. The Schubert calculus can be used to obtain lower bounds for certain real enumerative geometry problems (Schubert problems).

Odd-dimensional orbifolds with all geodesics closed are covered by manifolds

Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a characterization of orientable manifolds among orientable orbifolds in terms of characteristic classes.

Étale Steenrod operations and the Artin–Tate pairing

We prove a 1966 conjecture of Tate concerning the Artin–Tate pairing on the Brauer group of a surface over a finite field, which is the analog of the Cassels–Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen–Stoll on the Cassels–Tate pairing. Our method is based on studying a connection between the Artin–Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant étale Steenrod operations in terms of characteristic classes.

Characteristic classes and Hilbert–Poincaré series for perverse sheaves on abelian varieties

The convolution powers of a perverse sheaf on an abelian variety define an interesting family of branched local systems whose geometry is still poorly understood. We show that the generating series for their generic rank is a rational function of a very simple shape and that a similar result holds for the symmetric convolution powers. We also give formulae for other Schur functors in terms of characteristic classes on the dual abelian variety, and as an example we discuss the case of Prym-Tjurin varieties.

Indices of quaternionic complexes

Methods of parabolic geometries have been recently used to construct a class of elliptic complexes on quaternionic manifolds, Salamon's complex being the simplest case. The purpose of this paper is to describe an algorithm to compute their analytical indices in terms of characteristic classes. Using this, we are able to derive some topological obstructions to the existence of quaternionic structures on manifolds.

Obstructions to the existence of fold maps

We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in terms of characteristic classes, which arise as Postnikov invariants, and can be interpreted as primary and secondary obstructions to the elimination of certain singularities. We also discuss the relationship between the existence problem of fold maps and that of vector fields of stabilized tangent bundles.

Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.

The local defect index up to finite ambiguity

The rational Hopf invariant of the primary obstruction to a nowhere vanishing section in a vectorbundle is computed in terms of characteristic classes. This leads to a local rigidity result on the local defect index of a map.

On complex structures¶in 8-dimensional vector bundles

Necessary and sufficient conditions for the existence of a complex structure in an 8-dimensional spin vector bundle over a closed connected spin manifold of dimension 8 are given in terms of characteristic classes. The result completes the papers by Heaps [H] and Thomas [T] on the same topic.

On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes

Let ξ be an oriented 8-dimensional spin vector bundle over an 8-complex. Necessary and sufficient conditions for ξ to have 4 linearly independent sections or to be a sum of two 4-dimensional spin vector bundles are given in terms of characteristic classes and higher order cohomology operations. On closed connected spin smooth 8-manifolds these operations can be computed.

On $Sp(2)$ and $sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles

Let $\xi$ be an oriented 8-dimensional vector bundle. We prove that the structure group $SO(8)$ of $\xi$ can be reduced to $Sp(2)$ or $Sp(2)\cdot Sp(1)$ if and only if the vector bundle associated to $\xi$ via a certain outer automorphism of the group $Spin(8)$ has 3 linearly independent sections or contains a $3$-dimensional subbundle. Necessary and sufficient conditions for the existence of an $Sp(2)$-structure in $\xi$ over a closed connected spin manifold of dimension~$8$ are also given in terms of characteristic classes.

The cohomology of 𝐵𝑆𝑂_{𝑛} and 𝐵𝑂_{𝑛} with integer coefficients

The cohomology rings of B S O n BS{O_n} and B O n B{O_n} , with integer coefficients, are given by generators and relations expressed in terms of characteristic classes.

The classification of orientable vector bundles over CW-complexes of small dimension

SynopsisThe classification of orientable vector bundles over CW-complexes of dimension ≦8 is given in terms of characteristic classes using elementary homotopy theoretic methods and relations among characteristic classes.

THE INDEX OF RANDOM ELLIPTIC OPERATORS. I

This work is devoted to the construction of a theory of random elliptic operators; formulas are given for calculating the index of such an operator in terms of characteristic classes and the symbol of the operator. Bibliography: 15 titles.

Vector cross products on manifolds

possesses an almost complex structure. The properties of these manifolds have been investigated by Calabi [6], Gray [11], and Yano and Sumitomo [22]. The almost complex structure, which is a generalization of the one on S6, is defined in terms of a vector cross product on R7, which is a generalization of the ordinary Gibbs vector cross product on R3. In this paper we give a general definition of vector cross product (?2), and then study vector cross products on manifolds. Vector cross products are interesting for three reasons: first, they are themselves natural generalizations of the notion of almost complex structure; secondly, a vector cross product on a manifold M gives rise to unusual almost complex structures on certain submanifolds of M; thirdly, vector cross products provide one approach to the study of Riemannian manifolds with holonomy group G2 or Spin (7). Vector cross ptoducts on vector spaces have been studied from an algebraic standpoint in Brown and Gray [5] and from a topological standpoint in Eckmann [9] and Whitehead [20]. We consider the topological existence of vector cross products on manifolds in ?2. Then in ?3 we give conditions for the existence of vector cross products on vector bundles over CW-complexes in terms of characteristic classes. As an extra feature of our investigations we determine the behavior of the triality automorphism of Spin (8) on Spin characteristic classes. In ?4 we discuss vector cross products from a differential geometric point of view and in ?5 we develop some important relations between vector cross products and curvatures. The rest of the paper (?6 and ?7) is devoted to almost complex structures on manifolds. We show in ?6 that every orientable 6-dimensional submanifold of R8 possesses an almost complex structure. Such an almost complex structure is defined by means of a 3-fold vector cross product on R8. We also show that all of Calabi's manifolds can be obtained by this method. However, the almost complex structures constructed here are more general in several ways. In the first place, we consider 6-dimensional orientable submanifolds M of arbitrary 8-dimensional pseudo-Riemannian manifolds M possessing 3-fold vector cross products, e.g., parallelizable 8-dimensional manifolds. Secondly, the metric of M we use may be indefinite provided it has signature (4, 4). (Here we

Quantising proper actions on $\mathrm{Spin}^c$-manifolds

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to Spin$^c$-manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The result for cocompact actions is stated in terms of $K$-theory of group $C^*$-algebras, and the result for non-cocompact actions is an equality of numerical indices. In the non-cocompact case, the result generalises to Spin$^c$-Dirac operators twisted by vector bundles. This yields an index formula for Braverman's analytic index of such operators, in terms of characteristic classes on reduced spaces.