Related papers: Multiplicative properties of Morin maps

We introduce the notions of $\mathbb{K}$-framings, based $\mathbb{K}$-framings and relative $\mathbb{K}$-framings of a compact connected oriented surface $\Sigma$ for any commutative ring $\mathbb{K}$ with unit, and a map which maps a based…

By the classical result of Milnor and Novikov, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: $\Omega^U_*\simeq \mathbb Z[a_1,a_2,\dots]$, ${\rm deg}(a_i)=2i$. In this paper we solve a…

The cobordism ring of symplectic manifolds defined by V.L. Ginzburg is shown to be isomorphic to the Pontrjagin ring of complex-oriented manifolds with free circle actions. This suggests an interpretation of the formal group law of complex…

The purpose of this note is to relate certain ring-theoretic properties of rings in mixed and positive characteristics that are related to each other by a tilting operation used in perfectoid geometry. To this aim, we exploit the…

In this paper we show how the theory of monads can be used to deduce in a uniform manner several duality theorems involving categories of relations on one side and categories of algebras with homomorphisms preserving only some operations on…

For any ring R the category of monomorphisms is a full subcategory of the morphsim category over R, where the latter is equivalent to the module category of the triangular matrix ring with entries the ring R. In this work, we consider the…

Let X be a smooth simplicial toric variety. Let Z be the set of T-fixed points of X. We construct a filtration for A(Z), the ring of complex-valued functions on Z, such that Gr A(Z) is isomorphic to the cohomology algebra of X. This is the…

The moduli space of projective structures on a compact oriented surface $\Sigma$ has a holomorphic symplectic structure, which is constructed by pulling back, using the monodromy map, the Atiyah--Bott--Goldman symplectic form on the…

We study symplectic properties of monodromy map for second order linear equation with meromorphic potential having only simple poles on a Riemann surface. We show that the canonical symplectic structure on the cotangent bundle $T^*M_{g,n}$…

Let $p$ be a prime, let $d \geq 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P, Q \in A[x_1, \dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that…

We compute the rational cohomology ring of \bar R_2, the (compactified) moduli space of Prym curves of genus 2. We also recompute the rational cohomology ring of \bar S_2, the moduli space of spin curves of genus 2, thereby correcting some…

In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the…

We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are…

Let $K$ be a finite group and let $G$ be a finite group acting on $K$ by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures…

We establish a structure theorem for the connected automorphism groups of smooth complete toroidal horospherical varieties, that is, toric fibrations over rational homogeneous spaces. The key ingredient is a characterization of the Demazure…

We compute the quantum cohomology ring $H^*_{\varphi}({\bf P}, {\bf C})$ of an arbitrary $d$-dimensional smooth projective toric manifold ${\bf P}_{\Sigma}$ associated with a fan $\Sigma$. The multiplicative structure of $H^*_{\varphi}({\bf…

We use a theorem of Tolman and Weitsman to find explicit formul\ae for the rational cohomology rings of the symplectic reduction of flag varieties in C^n, or generic coadjoint orbits of SU(n), by (maximal) torus actions. We also calculate…

The problem of computing the integral cohomology ring of the symmetric square of a topological space has been of interest since the 1930s, but limited progress has been made on the general case until recently. In this work we offer a…

We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any…

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism…