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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
$f_{\ast }$
.
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 consider smooth, complex quasiprojective varieties
$U$
that admit a compactification with a boundary, which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on
$U$
vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.
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.
The ‘square peg problem’ or ‘inscribed square problem’ of Toeplitz asks if every simple closed curve in the plane inscribes a (nondegenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ‘homological’ nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper, we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs
$f$
,
$g:[t_{0},t_{1}]\rightarrow \mathbb{R}$
that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums
$y_{1}+y_{2}+y_{3}$
for
$y_{1},y_{2},y_{3}$
ranging in finite sets of real numbers.
We develop a theory of
$R$
-module Thom spectra for a commutative symmetric ring spectrum
$R$
and we analyze their multiplicative properties. As an interesting source of examples, we show that
$R$
-algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on
$R$
. We apply the general theory to obtain a description of the
$R$
-based topological Hochschild homology associated to an
$R$
-algebra Thom spectrum.
We construct combinatorial bases of the
$T$
-equivariant cohomology
$H_{T}^{\bullet }(\unicode[STIX]{x1D6F4},k)$
of the Bott–Samelson variety
$\unicode[STIX]{x1D6F4}$
under some mild restrictions on the field of coefficients
$k$
. These bases allow us to prove the surjectivity of the restrictions
$H_{T}^{\bullet }(\unicode[STIX]{x1D6F4},k)\rightarrow H_{T}^{\bullet }(\unicode[STIX]{x1D70B}^{-1}(x),k)$
and
$H_{T}^{\bullet }(\unicode[STIX]{x1D6F4},k)\rightarrow H_{T}^{\bullet }(\unicode[STIX]{x1D6F4}\setminus \unicode[STIX]{x1D70B}^{-1}(x),k)$
, where
$\unicode[STIX]{x1D70B}:\unicode[STIX]{x1D6F4}\rightarrow G/B$
is the canonical resolution. In fact, we also construct bases of the targets of these restrictions by picking up certain subsets of certain bases of
$H_{T}^{\bullet }(\unicode[STIX]{x1D6F4},k)$
and restricting them to
$\unicode[STIX]{x1D70B}^{-1}(x)$
or
$\unicode[STIX]{x1D6F4}\setminus \unicode[STIX]{x1D70B}^{-1}(x)$
respectively. As an application, we calculate the cohomology of the costalk-to-stalk embedding for the direct image
$\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{k}_{_{\unicode[STIX]{x1D6F4}}}$
. This algorithm avoids division by 2, which allows us to re-establish 2-torsion for parity sheaves in Braden’s example, Braden and Williamson [‘Modular intersection cohomology complexes on flag varieties’, Math. Z.272(3–4) (2012), 697–727].
We prove that the
$p$
-completed Brown–Peterson spectrum is a retract of a product of Morava
$E$
-theory spectra. As a consequence, we generalize results of Kashiwabara and of Ravenel, Wilson and Yagita from spaces to spectra and deduce that the notion of a good group is determined by Brown–Peterson cohomology. Furthermore, we show that rational factorizations of the Morava
$E$
-theory of certain finite groups hold integrally up to bounded torsion with height-independent exponent, thereby lifting these factorizations to the rationalized Brown–Peterson cohomology of such groups.
We study cup products in the integral cohomology of the Hilbert scheme of
$n$
points on a K3 surface and present a computer program for this purpose. In particular, we deal with the question of which classes can be represented by products of lower degrees.
We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further, we show that the images of the mapping class groups have non-trivial 2-cohomology, at least for small levels. For this purpose, we considered a series of quasi-homomorphisms on mapping class groups extending the previous work of Barge and Ghys (Math. Ann.294 (1992), 235–265) and of Gambaudo and Ghys (Bull. Soc. Math. France133(4) (2005), 541–579). These quasi-homomorphisms are pull-backs of the Dupont–Guichardet–Wigner quasi-homomorphisms on pseudo-unitary groups along quantum representations.
In this article we construct symmetric operations for all primes (previously known only for
$p=2$
). These unstable operations are more subtle than the Landweber–Novikov operations, and encode all
$p$
-primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map
$\mathbb{L}{\hookrightarrow}\mathbb{Z}[b_{1},b_{2},\ldots ]$
, providing an important structure on algebraic cobordism. Applications include questions of rationality of Chow group elements, and the structure of the algebraic cobordism. We also construct Steenrod operations of tom Dieck style in algebraic cobordism. These unstable multiplicative operations are more canonical and subtle than Quillen-style operations, and complement the latter.
We show that the
$\mathbb{Z}$
/2-equivariant nth integral Morava K-theory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in integral Morava K-theory with reality, and we recover the self-duality of the spectrum KO as a corollary. The study of
$\mathbb{Z}$
/2-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries of RO(
$\mathbb{Z}$
/2)-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.