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Homotopy theory folklore tells us that the sheaf defining the cohomology theory
$\operatorname {\mathrm {Tmf}}$
of topological modular forms is unique up to homotopy. Here we provide a proof of this fact, although we claim no originality for the statement. This retroactively reconciles all previous constructions of
$\operatorname {\mathrm {Tmf}}$
.
We show that every orbispace satisfying certain mild hypotheses has ‘enough’ vector bundles. It follows that the $K$-theory of finite rank vector bundles on such orbispaces is a cohomology theory. Global presentation results for smooth orbifolds and derived smooth orbifolds also follow.
For a weight structure w on a triangulated category
$\underline {C}$
we prove that the corresponding weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate objects’; this improves earlier conservativity formulations. In the case
$w=w^{sph}$
(the spherical weight structure on
$SH$
), we deduce the following converse to the stable Hurewicz theorem:
$H^{sing}_{i}(M)=\{0\}$
for all
$i<0$
if and only if
$M\in SH$
is an extension of a connective spectrum by an acyclic one. We also prove an equivariant version of this statement.
The main idea is to study M that has no weights
$m,\dots ,n$
(‘in the middle’). For
$w=w^{sph}$
, this is the case if there exists a distinguished triangle
$LM\to M\to RM$
, where
$RM$
is an n-connected spectrum and
$LM$
is an
$m-1$
-skeleton (of M) in the sense of Margolis’s definition; this happens whenever
$H^{sing}_i(M)=\{0\}$
for
$m\le i\le n$
and
$H^{sing}_{m-1}(M)$
is a free abelian group. We also consider morphisms that kill weights
$m,\dots ,n$
; those ‘send n-w-skeleta into
$m-1$
-w-skeleta’.
We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.
We study the
$E_2$
-algebra
$\Lambda \mathfrak {M}_{*,1}:= \coprod _{g\geqslant 0}\Lambda \mathfrak {M}_{g,1}$
consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion
$\Omega B\Lambda \mathfrak {M}_{*,1}$
: it is the product of
$\Omega ^{\infty }\mathbf {MTSO}(2)$
with a certain free
$\Omega ^{\infty }$
-space depending on the family of all boundary-irreducible mapping classes in all mapping class groups
$\Gamma _{g,n}$
with
$g\geqslant 0$
and
$n\geqslant 1$
.
Ganea proved that the loop space of
$\mathbb{C} P^n$
is homotopy commutative if and only if
$n=3$
. We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but
$\mathbb{C} P^3$
are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds
$G/T$
for a maximal torus T of a compact, connected Lie group G.
In this paper, we compute the $BP$-cohomology of complex projective Stiefel manifolds. The method involves the homotopy fixed point spectral sequence, and works for complex oriented cohomology theories. We also use these calculations and $BP$-operations to prove new results about equivariant maps between Stiefel manifolds.
For a subring $R$ of the rational numbers, we study $R$-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in ${\mathbb {A}}^1$-homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in ${\mathbb {A}}^1$-homotopy theory, paying attention to future applications for vector bundles. We show that $R$-localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $BGL_n$ is ${\mathbb {A}}^1$-nilpotent when $n$ is odd, and analyze the (more complicated) situation where $n$ is even as well. We establish analogs of various classical results about rationalization in the context of ${\mathbb {A}}^1$-homotopy theory: if $-1$ is a sum of squares in the base field, ${\mathbb {A}}^n \,{\setminus}\, 0$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres.
For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers and finite fields. We use this to extend Morel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck–Witt ring of quadratic forms to deeper base schemes.
We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.
Let $G$ be a finite group with cyclic Sylow $p$-subgroups, and let $k$ be a field of characteristic $p$. Then $H^{*}(BG;k)$ and $H_*(\Omega BG{{}^{{}^{\wedge }}_p};k)$ are $A_\infty$ algebras whose structure we determine up to quasi-isomorphism.
Let
$\mathscr {A}$
be a topological Azumaya algebra of degree
$mn$
over a CW complex X. We give conditions for the positive integers m and n, and the space X so that
$\mathscr {A}$
can be decomposed as the tensor product of topological Azumaya algebras of degrees m and n. Then we prove that if
$m<n$
and the dimension of X is higher than
$2m+1$
,
$\mathscr {A}$
may not have such decomposition.
For the $p$-localized sphere $\mathbb {S}^{2m-1}_{(p)}$ with $p >3$ a prime, we prove that the homotopy nilpotency satisfies $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}<\infty$, with respect to any homotopy associative $H$-structure on $\mathbb {S}^{2m-1}_{(p)}$. We also prove that $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}= 1$ for all but a finite number of primes $p >3$. Then, for the loop space of the associated $\mathbb {S}^{2m-1}_{(p)}$-projective space $\mathbb {S}^{2m-1}_{(p)}P(n-1)$, with $m,n\ge 2$ and $m\mid p-1$, we derive that $\mbox {nil}\ \Omega (\mathbb {S}^{2m-1}_{(p)}P (n-1))\le 3$.
In this paper we develop methods for classifying Baker, Richter, and Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss–Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on $\infty$-categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra $E$, we find that the ‘algebraic’ Azumaya algebras whose coefficient ring is projective are governed by the Brauer–Wall group of $\pi _0(E)$, recovering a result of Baker, Richter, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two Morita equivalence classes, depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum $KU$ are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization $KU[1/2]$ is $\mathbb {Z}/8\times \mathbb {Z}/2$. Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum $KO$ which become Morita-trivial $KU$-algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an ‘exotic’ $KO$-algebra with the same coefficient ring as $\mathrm {End}_{KO}(KU)$. This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of $KU$, previously studied by Mathew and Stojanoska.
Let X be a 4-dimensional toric orbifold. If $H^{3}(X)$ has a non-trivial odd primary torsion, then we show that X is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.
Let
$M(A,n)$
be the Moore space of type
$(A,n)$
for an Abelian group A and
$n\ge 2$
. We show that the loop space
$\Omega (M(A,n))$
is homotopy nilpotent if and only if A is a subgroup of the additive group
$\mathbb {Q}$
of the field of rationals. Homotopy nilpotency of loop spaces
$\Omega (M(A,1))$
is discussed as well.
We discuss some general properties of $\mathrm {TR}$ and its $K(1)$-localization. We prove that after $K(1)$-localization, $\mathrm {TR}$ of $H\mathbb {Z}$-algebras is a truncating invariant in the Land–Tamme sense, and deduce $h$-descent results. We show that for regular rings in mixed characteristic, $\mathrm {TR}$ is asymptotically $K(1)$-local, extending results of Hesselholt and Madsen. As an application of these methods and recent advances in the theory of cyclotomic spectra, we construct an analog of Thomason's spectral sequence relating $K(1)$-local $K$-theory and étale cohomology for $K(1)$-local $\mathrm {TR}$.
Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$-equivariant homology
$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of
$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $, making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].
We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq ~2$ we associate to an annular link $L$ a naive $\mathbb {Z}/r\mathbb {Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.