On invariants of foliated sphere bundles

The Hamilton–Jacobi characteristic equations for topological invariants: Pontryagin and Euler classes

On the cohomology of a polarization which contains the generators of a free torus action

Faddeev–Jackiw quantization of topological invariants: Euler and Pontryagin classes

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.

Free torus action on the manifold and the group of projectivities of a polytope P

We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.

Every closed Alexandrov space with a lower and upper curvature bound (in the triangle comparison sense) is a space of bounded curvature (in the sense of Berestovskii and Nikolaev). These spaces are topological manifolds and admit a canonical smooth structure such that, in its charts, the metric is induced by a Riemannian metric tensor with low regularity. In this note, we show that there are finitely many diffeomorphism types of closed, simply connected n-dimensional Alexandrov spaces with bounded curvature and upper diameter bound, provided $$2\le n\le 6$$. We also show that an analogous diffeomorphism finiteness result for such spaces does actually hold for general dimension n if, in addition, the second homotopy group is finite. Moreover, we prove that all closed, simply connected n-dimensional Alexandrov spaces with bounded curvature and upper diameter bound can be realized as quotients of finitely many closed, simply connected smooth manifolds with finite second homotopy group by some free torus action. These results extend well-known finiteness and realization theorems in Riemannian geometry to the more general setting of Alexandrov geometry. Indeed, our results may be simply viewed as concrete instances of the more general phenomenon that virtually any result that holds for a given class of closed Riemannian manifolds with bounded curvature and diameter may be extended to the Lipschitz closure of this class.

On the localization theorem at the cochain level and free torus actions

Minimal models of chain complexes associated with free torus actions on spaces have been extensively studied in the literature. In this paper, we discuss these constructions using the language of operads. The main goal of this paper is to define a new Koszul operad that has projections onto several of the operads used in these minimal model constructions.

It is known that in the integral cohomology of B \mathrm {SO}(2m) , the square of the Euler class is the same as the Pontryagin class in degree 4m . Given that the Pontryagin classes extend rationally to the cohomology of B STOP( 2m ), it is reasonable to ask whether the same relation between the Euler class and the Pontryagin class in degree 4m is still valid in the rational cohomology of B STOP( 2m ). In this paper we reformulate the hypothesis as a statement in differential topology, and also in a functor calculus setting.

0. Introduction. This paper is based on the simple observation that every 4-plane bundle over S4 admits a natural action of 50(3) by bundle maps and that every 8-plane bundle over 58 admits a natural action of the compact Lie group G2 by bundle maps. (These actions are easy to see once one remembers that SO (3) is the group of automorphisms of the quaternions and that G2 is the group of automorphisms of the Cayley numbers.) The study of these actions is closely connected to several well-known phenomena in differential topology and compact transformation groups. The most obvious connection is to Milnor's original construction of exotic 7-spheres as 3-sphere bundles over 54, [26]. Milnor proved that if the Euler class of such a bundle is a generator of H4(S4; Z), then its total space is homeomorphic to 57. He also defined a numerical invariant of the diffeomorphism type and used it to detect an exotic differential structure on some of these sphere bundles. Subsequently, Eells and Kuiper [12] introduced a refinement of this invariant, called the it-invariant. Using the ,u-invariant, they proved that the sphere bundle M7 (Milnor's notation) is a generator for the group of homotopy 7-spheres. These constructions also work for 7-sphere bundles over 58, [32], and the manifold M15 is a generator for bP16, the group of homotopy 15-spheres which bound wN-manifolds. It is a routine matter to check that Milnor's arguments and their subsequent refinements work G-equivariantly where G 5SO(3) or G2, and we shall do this in Sections 2 and 3. In particular, each sphere bundle with the correct Euler class is G-homeomorphic to an orthogonal action on S2,,+1, where 2n + 1 = 7 or 15. Moreover, distinct sphere bundles have distinct oriented G-diffeomorphism types. The proof of the second fact uses an equivariant version of the it-invariant. As originally defined this invariant takes values in Q/Z. However, it is well-known that in the presence of a G-action, with S C G, its value in Q is well-defined. Using

On Dold-Whitney's parallelizability of 4-manifolds

Class numbers, the novikov conjecture, and transformation groups

Curvature and torsion are the two tensors characterizing a general Riemannian space–time. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space–time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying space–time is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.

Curvature and torsion are the two tensors characterizing a general Riemannian space–time. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space–time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying space–time is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.