Mathematics

We prove a conjecture of Shklyarov concerning the relationship between K. Saito's higher residue pairing and a certain pairing on the periodic cyclic homology of matrix factorization categories. Along the way, we give new proofs of a result of Shklyarov and Polishchuk-Vaintrob's Hirzebruch-Riemann-Roch formula for matrix factorizations.

Applying recent results by Lowen-Van den Bergh we show that Hochschild cohomology is preserved under Koszul-Moore duality as a Gerstenhaber algebra. More precisely, the corresponding Hochschild complexes are linked by a quasi-isomorphism of B-infinity-algebras.

Assuming the classical Farrell-Jones conjecture we produce an explicit (commutative) group ring R and a thick subcategory C of perfect R -complexes such that the Waldhausen K -theory space K(C) is equivalent to a rational Eilenberg-Maclane space.

Using the free graded Lie algebras we introduce a natural subcomlex of the Loday's complex of a Leibniz algebra. Our conjecture says, that for free Leibniz algebras, the complex is acyclic.

In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define K -groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete Φ or edge metrics, can be regarded as the index pairing over these K -groups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.

This paper studies controlled operator K-theory on coarse spaces in light of new studies in arXiv:1710.06725 [math.AG].

In this paper, we construct an equivariant coarse homology theory with values in the category of non-commutative motives of Blumberg, Gepner and Tabuada, with coefficients in any small additive category. Equivariant coarse K-theory is obtained from the latter by passing to global sections. The present construction extends joint work of the first named author with Engel, Kasprowski and Winges by promoting codomain of the equivariant coarse K-homology functor to non-commutative motives.

We establish a variant of the splitting principle of Garibaldi-Merkurjev-Serre for invariants taking values in a strictly homotopy invariant sheaf. As an application, we prove the folklore result of Morel that pi_0 of the motivic localization of the group completion of the stack of finite étale schemes is given by the sheaf of unramified Grothendieck-Witt groups.

We equate various Euler classes of algebraic vector bundles, including those of [BM, KW, DJK], and one suggested by M.J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class, and give formulas for local indices at isolated zeros, both in terms of 6-functor formalism of coherent sheaves and as an explicit recipe in commutative algebra of Scheja and Storch. As an application, we compute the Euler classes associated to arithmetic counts of d-planes on complete intersections in P^n in terms of topological Euler numbers over R and C.

We apply the techniques developed by I. Panin for the proof of the equicharacteristic case of the Serre-Grothendieck conjecture for isotropic reductive groups (I. Panin, A. Stavrova, N. Vavilov, 2015; I. Panin, 2019) to obtain similar injectivity and A^1-invariance theorems for non-stable K_1-functors associated to isotropic reductive groups. Namely, let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We show that if G has isotropic rank >=2 and R is a regular domain containing a field, then K_1^G(R[x])=K_1^G(R) for any n>=1, where K_1^G(R)=G(R)/E(R) is the corresponding non-stable K_1-functor, also called the Whitehead group of G. If R is, moreover, local, then we show that K_1^G(R)->K_1^G(K) is injective, where K is the field of fractions of R.