We observe that the Poincaré duality isomorphism for a string manifold is an isomorphism of modules over the subalgebra A(2) of the modulo 2 Steenrod algebra. In particular, the pattern of the operations Sq, Sq, and Sq on the cohomology of a string manifold has a symmetry around the middle dimension. We characterize this kind of cohomology operation duality in term of the annihilator of the Tho...

We construct a canonical Thom isomorphism for virtual T-oriented T-equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the rational case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the re...

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M . This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric constr...

The mould Penicillium frequentans Westling is, as its name implies, of very common occurrence on all kinds of material. According to Raper & Thom (1949), the Monoverticillate section of the genus Penicillium includes a P. frequentans series of three species, namely the type species, P. sprnulosum Thom and a less common species, P. purpurrescens (Sopp) n.comb. Webmer's classical studies of citri...

We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor Σ∞+ Ω ∞ : ho(connective spectra) → ho(E∞ ring spectra). To a map of spectra f : b → bgl1A, we associate an E∞ A-algebra Thom spectrum Mf , which...

We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of Legendrian characteristic classes such that any Legendrian Thom polynomial has nonnegative coefficients when expanded in these bases. The method uses a suitable Lagrange Grassmann bundle on the product of projective spaces. This is an extension of a nonnegativity resul...