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We calculate the integral equivariant cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree ordinary cohomology vanishes. We begin by studying GKM-orbifolds, which are more general, before specializing to half-dimensional torus actions.
We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on p-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of p-adic simplicial volumes. As the main examples, we compute the weightless and p-adic simplicial volumes of surfaces. This is based on an alternative way to calculate classical simplicial volume of surfaces without hyperbolic straightening and shows that surfaces satisfy mod p and p-adic approximation of simplicial volume.
We give a geometric interpretation of sheaf cohomology for higher degrees
$n\geq 1$
in terms of torsors on the member of degree
$d=n-1$
in hypercoverings of type
$r=n-2$
, endowed with an additional datum, the so-called rigidification. This generalizes the fact that cohomology in degree one is the group of isomorphism classes of torsors, where the rigidification becomes vacuous, and that cohomology in degree two can be expressed in terms of bundle gerbes, where the rigidification becomes an associativity constraint.
We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of
$\mathbf{SL}_{n}(\mathbb{Z})$
. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group
$\mathbf{SL}_{n}(K)$
for
$K$
a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group
$\mathbf{SL}_{n}(\mathbb{Z})$
.
The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a generalized n-manifold Xn, in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the nth Steenrod homology group $H^{st}_{n} (X^{n}, \mathbb {L}^+)$, where 𝕃+ is the connected covering spectrum of the periodic surgery spectrum 𝕃, avoiding the use of the geometric splitting procedure, the use of which is standard in surgery on topological manifolds.
Let
$X$
be a topological space. We consider certain generalized configuration spaces of points on
$X$
, obtained from the cartesian product
$X^{n}$
by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call ‘twisted commutative dg algebra models’ for the cochains on
$X$
. Suppose that
$X$
is a ‘nice’ topological space,
$R$
is any commutative ring,
$H_{c}^{\bullet }(X,R)\rightarrow H^{\bullet }(X,R)$
is the zero map, and that
$H_{c}^{\bullet }(X,R)$
is a projective
$R$
-module. We prove that the compact support cohomology of any generalized configuration space of points on
$X$
depends only on the graded
$R$
-module
$H_{c}^{\bullet }(X,R)$
. This generalizes a theorem of Arabia.
We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle–Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimension at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, including certain manifolds with non-trivial higher homotopy groups and certain products of aspherical manifolds.
For a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.
This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.
We study the equivariant oriented cohomology ring
$\mathtt{h}_{T}(G/P)$
of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in
$\mathtt{h}_{T}(G/P)$
can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory
$\mathfrak{h}$
corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in
$\mathfrak{h}_{T}(G/P)$
. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.
The circle transfer
$Q\Sigma (LX_{hS^1})_+ \to QLX_+$
has appeared in several contexts in topology. In this note, we observe that this map admits a geometric re-interpretation as a morphism of cobordism categories of 0-manifolds and 1-cobordisms. Let 𝒞1(X) denote the one-dimensional cobordism category and let Circ(X) ⊂ 𝒞1(X) denote the subcategory whose objects are disjoint unions of unparametrized circles. Multiplication in S1 induces a functor Circ(X) → Circ(LX), and the composition of this functor with the inclusion of Circ(LX) into 𝒞1(LX) is homotopic to the circle transfer. As a corollary, we describe the inclusion of the subcategory of cylinders into the two-dimensional cobordism category 𝒞2(X) and find that it is null-homotopic when X is a point.
We establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.
Let
$X$
be a compact, metric and totally disconnected space and let
$f:X\rightarrow X$
be a continuous map. We relate the eigenvalues of
$f_{\ast }:\check{H}_{0}(X;\mathbb{C})\rightarrow \check{H}_{0}(X;\mathbb{C})$
to dynamical properties of
$f$
, roughly showing that if the dynamics is complicated then every complex number of modulus different from 0, 1 is an eigenvalue. This stands in contrast with a classical inequality of Manning that bounds the entropy of
$f$
below by the spectral radius of
$f_{\ast }$
.
Holmsen, Kynčl and Valculescu recently conjectured that if a finite set
$X$
with
$\ell n$
points in
$\mathbb{R}^{d}$
that is colored by
$m$
different colors can be partitioned into
$n$
subsets of
$\ell$
points each, such that each subset contains points of at least
$d$
different colors, then there exists such a partition of
$X$
with the additional property that the convex hulls of the
$n$
subsets are pairwise disjoint.
We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least
$c$
different colors, where we also allow
$c$
to be greater than
$d$
. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from
$c$
different colors. For example, when
$n\geqslant 2$
,
$d\geqslant 2$
,
$c\geqslant d$
with
$m\geqslant n(c-d)+d$
are integers, and
$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$
are
$m$
positive finite absolutely continuous measures on
$\mathbb{R}^{d}$
, we prove that there exists a partition of
$\mathbb{R}^{d}$
into
$n$
convex pieces which equiparts the measures
$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{d-1}$
, and in addition every piece of the partition has positive measure with respect to at least
$c$
of the measures
$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$
.
Functorial semi-norms on singular homology give refined ‘size’ information on singular homology classes. A fundamental example is the ℓ1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the ℓ1-semi-norm.
We define and study twisted Alexander-type invariants of complex hypersurface complements. We investigate torsion properties for the twisted Alexander modules and extend the local-to-global divisibility results of Maxim and of Dimca and Libgober to the twisted setting. In the process, we also study the splitting fields containing the roots of the corresponding twisted Alexander polynomials.
We study the germs at the origin of
$G$
-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group
$G$
is either
$\text{SL}_{2}(\mathbb{C})$
or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either
$G=\text{SL}_{n}(\mathbb{C})$
for some
$n\geqslant 3$
, or the depth is greater than 1, then certain natural inclusions of germs are strict.
We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning the third differential of the spectral sequence, and recover known results when the twisting comes from finite order elements in discrete torsion.