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For certain pairs of Lie groups (G, H) and primes p, Harris showed a relation of the p-localized homotopy groups of G and H. This is reinterpreted as a p-local homotopy equivalence G ≃ (p)H × G/H, and so there is a projection G(p) → H(p). We show how much this projection preserves the higher homotopy associativity.
On établit une décomposition de l’homologie stable des groupes d’automorphismes des groupes libres à coefficients polynomiaux contravariants en termes d’homologie des foncteurs. Elle permet plusieurs calculs explicites, qui recoupent des résultats établis de manière indépendante par O. Randal-Williams et généralisent certains d’entre eux. Nos méthodes reposent sur l’examen d’extensions de Kan dérivées associées à plusieurs catégories de groupes libres, la généralisation d’un critère d’annulation homologique à coefficients polynomiaux dû à Scorichenko, le théorème de Galatius identifiant l’homologie stable des groupes d’automorphismes des groupes libres à celle des groupes symétriques, la machinerie des
$\unicode[STIX]{x1D6E4}$
-espaces et le scindement de Snaith.
We introduce a notion of Koszul A∞-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A∞-algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.
The Lusternik–Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik–Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product
$G\times \mathbb{R}^{n}$
and apply the results to the planar and spatial motion of two rigid bodies in
$\mathbb{R}^{2}$
and
$\mathbb{R}^{3}$
respectively.
For a path connected space X, the homology algebra
$H_*(QX; \mathbb{Z}/2)$
is a polynomial algebra over certain generators QIx. We reinterpret a technical observation, of Curtis and Wellington, on the action of the Steenrod algebra A on the Λ algebra in our terms. We then introduce a partial order on each grading of H*QX which allows us to separate terms in a useful way when computing the action of dual Steenrod operations
$Sq^i_*$
on
$H_*(QX; \mathbb{Z}/2)$
. We use these to completely characterise the A-annihilated generators of this polynomial algebra. We then propose a construction for sequences I so that QIx is A-annihilated. As an application, we offer some results on the form of potential spherical classes in H*QX upon some stability condition under homology suspension. Our computations provide new numerical conditions in the context of hit problem.
This paper concerns extension of maps using obstruction theory under a non-classical viewpoint. It is given a classification of homotopy classes of maps and as an application it is presented a simple proof of a theorem by Adachi about equivalence of vector bundles. Also it is proved that, under certain conditions, two embeddings are homotopic up to surgery if and only if the respective normal bundles are SO-equivalent.
An n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.
For almost any compact connected Lie group
$G$
and any field
$\mathbb{F}_{p}$
, we compute the Batalin–Vilkovisky algebra
$H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$
on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if
$p$
is odd or
$p=0$
, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology
$HH^{\star }(H_{\star }(G),H_{\star }(G))$
. Over
$\mathbb{F}_{2}$
, such an isomorphism of Batalin–Vilkovisky algebras does not hold when
$G=\text{SO}(3)$
or
$G=G_{2}$
. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.
It is well known that if X is a CW-complex, then for every weak homotopy equivalence f : A → B, the map f* : [X, A] → [X, B] induced in homotopy classes is a bijection. In fact, up to homotopy equivalence, only CW-complexes have that property. Now, for which spaces X is f* : [B, X] → [A, X] a bijection for every weak equivalence f? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.
If
$K$
is a simplicial complex on
$m$
vertices, the flagification of
$K$
is the minimal flag complex
$K^{f}$
on the same vertex set that contains
$K$
. Letting
$L$
be the set of vertices, there is a sequence of simplicial inclusions
$L\stackrel{}{\longrightarrow }K\stackrel{}{\longrightarrow }K^{f}$
. This induces a sequence of maps of polyhedral products
$(\text{}\underline{X},\text{}\underline{A})^{L}\stackrel{g}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K}\stackrel{f}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K^{f}}$
. We show that
$\unicode[STIX]{x1D6FA}f$
and
$\unicode[STIX]{x1D6FA}f\circ \unicode[STIX]{x1D6FA}g$
have right homotopy inverses and draw consequences. For a flag complex
$K$
the polyhedral product of the form
$(\text{}\underline{CY},\text{}\underline{Y})^{K}$
is a co-
$H$
-space if and only if the 1-skeleton of
$K$
is a chordal graph, and we deduce that the maps
$f$
and
$f\circ g$
have right homotopy inverses in this case.
Let M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M, ∂M) can be recovered from the configuration category of M \ ∂M. The grouplike monoid of derived homotopy automorphisms of the configuration category of M \ ∂M then acts on the homotopical model of (M, ∂M). That action is compatible with a better known homotopical action of the homeomorphism group of M \ ∂M on (M, ∂M).
For a prime
$p$
, let
$\hat{F}_{p}$
be a finitely generated free pro-
$p$
-group of rank at least
$2$
. We show that the second discrete homology group
$H_{2}(\hat{F}_{p},\mathbb{Z}/p)$
is an uncountable
$\mathbb{Z}/p$
-vector space. This answers a problem of A. K. Bousfield.
Let G be a simple, simply connected, compact Lie group, and let M be an orientable, smooth, connected, closed 4-manifold. In this paper, we calculate the homotopy type of the suspension of M and the homotopy types of the gauge groups of principal G-bundles over M when π1(M) is (1) ℤ*m, (2) ℤ/prℤ, or (3) ℤ*m*(*nj=1ℤ/prjjℤ), where p and the pj's are odd primes.
This paper sets up the foundations for derived algebraic geometry, Goerss–Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën–Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and
$\text{E}_{\infty }$
-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of
$\text{E}_{\infty }$
-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.
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 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 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.
Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real
$K$
-theory spectra of Hopkins and Miller at height
$n=p-1$
, for
$p$
an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra
$E_{n}^{hG}$
, where
$E_{n}$
is Lubin–Tate
$E$
-theory at the prime
$p$
and height
$n=p-1$
, and
$G$
is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension.
In this paper, we show that the known models for (∞, 1)-categories can all be extended to equivariant versions for any discrete group G. We show that in two of the models we can also consider actions of any simplicial group G.
In this note, we have obtained a Whitehead-like tower of a module by considering a suitable set of morphisms and shown that the different stages of the tower are the Adams cocompletions of the module with respect to the suitably chosen set of morphisms.