Torsion In Cohomology Research Articles

Torsion in cohomology and dimensional reduction

Conventional wisdom dictates that ℤN factors in the integral cohomology group Hp(Xn, ℤ) of a compact manifold Xn cannot be computed via smooth p-forms. We revisit this lore in light of the dimensional reduction of string theory on Xn, endowed with a G-structure metric that leads to a supersymmetric EFT. If massive p-form eigenmodes of the Laplacian enter the EFT, then torsion cycles coupling to them will have a non-trivial smeared delta form, that is an EFT long-wavelength description of p-form currents of the (n − p)-cycles of Xn. We conjecture that, whenever torsion cycles are calibrated, their linking number can be computed via their smeared delta forms. From the EFT viewpoint, a torsion factor in cohomology corresponds to a ℤN gauge symmetry realised by a Stückelberg-like action, and calibrated torsion cycles to BPS objects that source the massive fields involved in it.

Symmetry TFTs from String Theory

We determine the d+1 dimensional topological field theory, which encodes the higher-form symmetries and their ’t Hooft anomalies for d-dimensional QFTs obtained by compactifying M-theory on a non-compact space X. The resulting theory, which we call the Symmetry TFT, or SymTFT for short, is derived by reducing the topological sector of 11d supergravity on the boundary partial X of the space X. Central to this endeavour is a reformulation of supergravity in terms of differential cohomology, which allows the inclusion of torsion in cohomology of the space partial X, which in turn gives rise to the background fields for discrete (in particular higher-form) symmetries. We apply this framework to 7d super-Yang Mills, where X= mathbb {C}^2/Gamma _{ADE}, as well as the Sasaki–Einstein links of Calabi–Yau three-fold cones that give rise to 5d superconformal field theories. This M-theory analysis is complemented with a IIB 5-brane web approach, where we derive the SymTFTs from the asymptotics of the 5-brane webs. Our methods apply to both Lagrangian and non-Lagrangian theories, and allow for many generalisations.

Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps

On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel–Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsion in cohomology of arithmetic groups.

Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero

We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.

Torsion in the cohomology of torus orbifolds

The authors study torsion in the integral cohomology of a certain family of 2n-dimensional orbifolds X with actions of the n-dimensional compact torus. Compact simplicial toric varieties are in our family. For a prime number p, the authors find a necessary condition for the integral cohomology of X to have no p-torsion. Then it is proved that the necessary condition is sufficient in some cases. The authors also give an example of X which shows that the necessary condition is not sufficient in general.

SUR LA TORSION DANS LA COHOMOLOGIE DES VARIÉTÉS DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR

(Torsion in the cohomology of Kottwitz–Harris–Taylor Shimura varieties) When the level at $l$ of a Shimura variety of Kottwitz–Harris–Taylor is not maximal, its cohomology with coefficients in a $\overline{\mathbb{Z}}_{l}$-local system isn’t in general torsion free. In order to prove torsion freeness results of the cohomology, we localize at a maximal ideal $\mathfrak{m}$ of the Hecke algebra. We then prove a result of torsion freeness resting either on $\mathfrak{m}$ itself or on the Galois representation $\overline{\unicode[STIX]{x1D70C}}_{\mathfrak{m}}$ associated to it. Concerning the torsion, in a rather restricted case than Caraiani and Scholze (« On the generic part of the cohomology of compact unitary Shimura varieties », Preprint, 2015), we prove that the torsion doesn’t give new Satake parameters systems by showing that each torsion cohomology class can be raised in the free part of the cohomology of a Igusa variety.

A note on secondary K-theory

We prove that Toen’s secondary Grothendieck ring is isomorphic to the Grothendieck ring of smooth proper pretriangulated dg categories previously introduced by Bondal, Larsen, and Lunts. Along the way, we show that those short exact sequences of dg categories in which the first term is smooth proper and the second term is proper are necessarily split. As an application, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a commutative ring of characteristic zero, it distinguishes between dg Azumaya algebras associated to nontorsion cohomology classes and dg Azumaya algebras associated to torsion cohomology classes (= ordinary Azumaya algebras); in the case of a field of characteristic zero, it is injective; in the case of a field of positive characteristic p, it restricts to an injective map on the p-primary component of the Brauer group.

Non-Abelian discrete gauge symmetries in F-theory

The presence of non-Abelian discrete gauge symmetries in four-dimensional F-theory compactifications is investigated. Such symmetries are shown to arise from seven-brane configurations in genuine F-theory settings without a weak string coupling description. Gauge fields on mutually non-local seven-branes are argued to gauge both R-R and NS-NS two-form bulk axions. The gauging is completed into a generalisation of the Heisenberg group with either additional seven-brane gauge fields or R-R bulk gauge fields. The former case relies on having seven-brane fluxes, while the latter case requires torsion cohomology and is analysed in detail through the M-theory dual. Remarkably, the M-theory reduction yields an Abelian theory that becomes non-Abelian when translated into the correct duality frame to perform the F-theory limit. The reduction shows that the gauge coupling function depends on the gauged scalars and transforms non-trivially as required for the groups encountered. This field dependence agrees with the expectations for the kinetic mixing of seven-branes and is unchanged if the gaugings are absent.

Equivariant torsion and base change

What is true order of growth of torsion in of an arithmetic group? Let $D$ be a quaternion over an imaginary quadratic field $F.$ Let $E/F$ be a cyclic Galois extension with $\mathrm{Gal}(E/F) = \langle \sigma \rangle.$ We prove lower bounds for the Lefschetz number of $\sigma$ acting on torsion cohomology of certain Galois-stable arithmetic subgroups of $D_E^\times.$ For these same subgroups, we unconditionally prove a would-be-numerical consequence of existence of a hypothetical base change map for torsion cohomology.

Torsion cohomology for solvable groups of finite rank

We define a class U of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that G is a group in U and A a ZG-module. If A is Z-torsion-free and has finite Z-rank, we stipulate a condition on A that guarantees that Hn(G,A) and Hn(G,A) must be finite for n≥0. Moreover, if the underlying abelian group of A is a Černikov group, we identify a similar condition on A that ensures that Hn(G,A) must be a Černikov group for all n≥0.

THE EQUIVARIANT CHEEGER–MÜLLER THEOREM ON LOCALLY SYMMETRIC SPACES

In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion, and demonstrate that Bismut–Zhang’s equivariant Cheeger–Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and torsion cohomology comparison for arithmetic locally symmetric spaces to an equivariant setting.

The real section conjecture and Smith's fixed-point theorem for pro-spaces

We prove a topological version of the section conjecture for the profinite completion of the fundamental group of finite CW-complexes equipped with the action of a group of prime order $p$ whose $p$-torsion cohomology can be killed by finite covers. As an application we derive the section conjecture for the real points of a large class of varieties defined over the field of real numbers and the natural analogue of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers.

On the first group of the chromatic cohomology of graphs

The Hochschild homology of the algebra of truncated polynomials \({{\mathcal {A}_m=\mathbb {Z}[x]/(x^m)}}\) is closely related to the Khovanov-type homology as shown by the second author. In the present paper we utilize this in the study of the first graph cohomology group of an arbitrary graph G with v vertices. The complete description of this group is given for m = 2, 3. For the algebra \({\mathcal {A}_2}\) we relate the chromatic graph cohomology with the Khovanov homology of adequate links. We describe the chromatic cohomology over the algebra \({\mathcal {A}_3}\) using the homology of a cell complex built on the graph G. In particular we prove that \({{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}\) can be isomorphic to any finite abelian group. Moreover, we give a characterization of graphs which have torsion in cohomology \({{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}\) and construct graphs which have the same (di)chromatic polynomial but different \({{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}\) .

Torsion cohomology classes and algebraic cycles on complex projective manifolds

Atiyah and Hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold, which are not Poincaré dual to an algebraic cycle. We notice that the order of these classes must be small compared to the dimension of the manifold. However, building upon a construction of Kollár, one can provide such examples with arbitrary high prime order, the dimension being fixed. This method also provides examples of torsion algebraic cycles, which are non trivial in the Griffiths group, and lie in a arbitrary high level of the H. Saito filtration on Chow groups.

Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups

The purpose of the present note is to give a general procedure for calculating the singular cohomology H*(K, IFp) with coefficients in an arbitrary finite field IFv of a connected compact Lie group K. Simultaneously, we calculate the Chow ring A(G, IFp)..=A(G)| p of the corresponding complex algebraic group G, and the degrees of the basic "generalized invariants" of the Weyl group W in arbitrary characteristic p. A mysterious connection between the degrees of generators of A(G, IFp) and the kernel of the principal nilpotent element in characteristic p is pointed out. The computation of real cohomology of compact Lie groups started by E. Cartan in 1929 was completed for classical groups by R. Brauer and by L. Pontryagin in 1935. H. Hopf and H. Samelson showed in 1941 that H*(K, F,.) is an exterior algebra on generators of degrees 2 d 1 1 . . . . ,2dr-l , where l = r a n k K . But only in 1949 C.T. Yen managed to compute the d; for all exceptional groups, case by case. On the other hand, C. Chevalley and J. Leray showed that the d i are nothing else but the degrees of the basic invariants over of the Weyl group W. A simple procedure for calculating the dl was given in the late fifties by A.J. Coleman and by B. Kostant. Nowadays this is common knowledge. The problem of computing the torsion in H*(K,Z) proved to be more recalcitrant. This was done by several authors: C.E. Miller settled SO,, in 1953 [,23], A. Borel settled Spin,, Ad Spin,, and the quotients of SU, and Sp, in 1954 [,6], P.F. Baum and W. Browder settled the remaining classical group, the semispin group, in 1965 [-4]. G 2 and F 4 were settled in 1954 [6], and the p > 3 torsion for E 6 and p>5tors ion for Ev,E s in 1961 [,7], by A. Borel. S. Araki computed the 2-torsion for E 6 , E 7 [-1] and the 3-torsion for E 7 , E s [2], and announced the 2-torsion for E 8 in a joint paper with Y. Shikata in 1961 [-3]. The details of the proof for the last result have appeared only recently [,18]. There was an error in [,1] in the case of Ad E 7, which has been fixed in [,13].