arXiv research

Geometrically interprets a duality theorem linking cochain and chain complexes.

New proof of chain duality for simplicial complexes.

We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincaré duality. The resulting double-cobordism groups are a refinement of Ranicki's torsion algebraic $L$-groups for localisations of a commutative ring with involution. The refinement is analogous to the difference between metabo…

Let M be an orientable topological manifold of dimension m, m greater or equal to 5, with fundamental group $Γ$. Let S(M) be the topological structure set, endowed with the group structure induced by its identification with Ranicki's algebraic structure set. We prove that the (rationalized) rho map $ρ_Γ: S(M)\rightarro…

We refine the Whitehead torsion of a chain equivalence of finite chain complexes in an additive category $\bA$ from an element of $\widetilde{K}^{iso}_1(\bA)$ to an element of the absolute group $K_1^{iso}(\bA)$. We apply this invariant to symmetric Poincaré complexes and identify it in terms of more traditional invari…

This paper reverses a construction by merging boundary critical points into an interior one.

We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding is at least two. This is a topological counterpart of the algebraic splitting theorem for embedded cobordisms of the first author, A. Nemethi…

New examples show some manifolds can't be decomposed.

The total surgery obstruction of a finite n-dimensional Poincare complex X is an element s(X) of a certain abelian group S_n (X) with the property that for n >= 5 we have s(X) = 0 if and only if X is homotopy equivalent to a closed n-dimensional topological manifold. The definitions of S_n (X) and s(X) and the property…

Let $n$ be a positive integer, and let $\ell>1$ be square-free odd. We classify the set of equivariant homeomorphism classes of free $C_\ell$-actions on the product $S^1 \times S^n$ of spheres, up to indeterminacy bounded in $\ell$. The description is expressed in terms of number theory. The techniques are various appl…

For a closed topological manifold M with dim (M) >= 5 the topological structure set S(M) admits an abelian group structure which may be identified with the algebraic structure group of M as defined by Ranicki. If dim (M) = 2d-1, M is oriented and M is equipped with a map to the classifying space of a finite group G, th…

The validity of Freedman's disk theorem is known to depend only on the fundamental group. It was conjectured that it fails for nonabelian free fundamental groups. If this were true then surgery theory would work in dimension four. Recently, Krushkal and Lee proved a surprising result that surgery theory works for a lar…

In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which …

Proves Poincaré duality for Hopf algebroids with bijective antipode.

Introduces Kähler duality between domains in complex space.

Research on dualities in geometric stereotypes.

Duality restored in gauge theory, gravity, and string theory models.

Unified proof of four Bavard dualities and new results on quasimorphisms.

Verma Howe duality connects tensor products of Verma modules to LKB representations.

Cohomological and homological spectral sequences are shown to be isomorphic.

The paper proves T-duality and Hori formulae for winding loop spaces.

Equivariant T-duality connects bundles with twists.

We give the definition of a duality that is applicable to arbitrary $k$-forms. The operator that defines the duality depends on a fixed form $Ω$. Our definition extends in a very natural way the Hodge duality of $n$-forms in $2n$ dimensional spaces and the generalized duality of two-forms. We discuss the properties of …

The paper establishes T-duality for 2D σ-models with H-flux.

QP perspective on Poisson-Lie T-duality topology changes.

We develop new algebraic methods refining the Witt group of linking forms and Ranicki's torsion algebraic L-groups into double Witt groups and double L-groups. At each prime ideal of the underlying ring, our double Witt groups capture infinitely many more integral signatures of the linking form than the single Witt gro…

Unified framework for T-duality in both trivial and non-trivial topologies.

Geometric duality connects graph isomorphism and knot equivalence.

Koszul duality for manifold modules proven.

This dissertation explores T-duality between hyperkähler structures and branes on algebraic integrable systems.

Maps self-duality in little disks operad to framed manifolds.

We study generalized complex structures and $T$-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called "Infinitesimal $T$-duality". As an application we deal with the problem of finding symplectic stru…

New spherical T-duality for higher degree forms in fiber bundles.

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group of the base acts trivially on the cohomology ring of the fibre with real coefficients, in which case the signature of the total space equals the product of the signatures of base and fibre. Hambl…

The paper establishes a duality between non-compact and compact symmetric pairs.

We find the T-duality transformation rules for 2-dimensional (2,1) supersymmetric sigma-models in (2,1) superspace. Our results clarify certain aspects of the (2,1) sigma model geometry relevant to the discussion of T-duality. The complexified duality transformations we find are equivalent to the usual Buscher duality …

We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X d…

The paper shows plentiful non-homotopy finite Poincaré duality spaces.

Study S-duality and supersymmetry on curved manifolds using localization.

We study the convex duality method for robust utility maximization in the presence of a random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true for a wide class of utility functions on the whole real line and unbounded random endo…

In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a $δ$-hyperbolic Poincare Duality group, then the fixed subgroup is a Poincare Duality group over Z/p. We also provide examples to show that the fixed subgroup might not even be a Duality group over Z.

Dualities in physics help in machine learning tasks.

New approach to T-duality using Courant algebroids.

Proves a theorem for 3D Poincaré duality pairs.

New Langlands duality conjectures for 3-manifold skein modules.

Explains partial duality for ribbon graphs in simple terms.

The paper proves a generalized Lefschetz duality for a specific type of manifold.

Extends T-duality to more general fibrations, proving isomorphic cohomology and algebroids.