Using the derived algebraic formalism of Lurie we describe a framework for coalgebraic structures which are neither necessarily commutative nor cocommutative. We show that Thom spectra and homotopical descent data are both natural examples of this kind of structure.

Combining the “method of restriction equations” of Rimányi et al. with the techniques of symmetric functions, we establish the Schur function expansions of the Thom polynomials for the Morin singularities A3 : (C •, 0) → (C, 0) for any nonnegative integer k.

We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin-Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in En-algebra.

More than half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word “generically”, this statement is ostensibly true and is nowadays commonly accepted. On the other hand, the (f...

From the viewpoint of algebraic topology, a quasitoric orbifold is a singular space that is constructed out of a simple polytope and a related integral matrix, and admits a well-behaved torus action. Weighted projective spaces P(χ) provide a particularly illuminating class of examples (where χ denotes a weight vector of natural numbers), but their topological literature is remarkably sparse. Ou...

whose domain is the symplectic bordism ring of V. L. Ginzburg [2]. As Ginzburg proved σ to be injective, this establishes it to be an isomorphism, a result first proved by J. Morava [3] by a more topological argument. We also make some observations on the associated homology theory. For the benefit of topologists, we remark that the notion of manifold with symplectic structure is distinct from ...