Dirac rings are commutative algebras in the symmetric monoidal category of $\mathbb {Z}$-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger $\infty $-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to $\operatorname {MU}$ and $\mathbb {F}_p$ in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves.

We introduce Chern classes in $U(m)$-equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the $\mathbf {MU}$-cohomology of $B U(m)$. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the $\mathbf {MU}$-completion theorem of Greenlees–May and La Vecchia.

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 prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

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 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.

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.

The algebraic cobordism group of a scheme is generated by cycles that are proper morphisms from smooth quasiprojective varieties. We prove that over a field of characteristic zero the quasiprojectivity assumption can be omitted to get the same theory.

We construct a cohomology theory using quasi-smooth derived schemes as generators and an analog of the bordism relation using derived fiber products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that, in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.

In 1969 Quillen discovered a deep connection between complex cobordism and formal group laws which he announced in [Qui69]. Algebraic topology has never been the same since. We will describe the content of [Qui69] and then discuss its impact on the field. This paper is a writeup of a talk on the same topic given at the Quillen Conference at MIT in October 2012. Slides for that talk are available on the author's home page.

We show that there is an essentially unique S-algebra structure on the Morava K-theory spectrum K(n), while K(n) has uncountably many MU or -algebra structures. Here is the K(n)-localized Johnson–Wilson spectrum. To prove this we set up a spectral sequence computing the homotopy groups of the moduli space of A∞ structures on a spectrum, and use the theory of S-algebra k-invariants for connectiveS-algebras found in the work of Dugger and Shipley [Postnikov extensions of ring spectra, Algebr. Geom. Topol. 6 (2006), 1785–1829 (electronic)] to show that all the uniqueness obstructions are hit by differentials.

The $H$-space that represents Brown-Peterson cohomology $\text{B}{{\text{P}}^{k}}\left( - \right)$ was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum $P\left( n \right)$, by constructing idempotent operations in $P\left( n \right)$-cohomology $P{{(n)}^{k}}\left( - \right)$ in the style of Boardman-Johnson-Wilson; this relies heavily on the Ravenel-Wilson determination of the relevant Hopf ring. The resulting $\left( i\,-\,1 \right)$-connected $H$-spaces ${{Y}_{i}}$ have free connective Morava $K$-homology $k{{(n)}_{*}}({{Y}_{i}})$, and may be built from the spaces in the $\Omega$-spectrum for $k\left( n \right)$ using only ${{v}_{n}}$-torsion invariants.

We also extend Quillen's theorem on complex cobordism to show that for any space $X$, the $P{{\left( n \right)}_{*}}$-module $P{{(n)}^{*}}\,(X)$ is generated by elements of $P{{(n)}^{i}}(X)$ for $i\,\ge \,0$. This result is essential for the work of Ravenel-Wilson-Yagita, which in many cases allows one to compute BP-cohomology from Morava $K$-theory.

We construct a spectral sequence that computes the generalized homology E*(∏ Xα) of a product of spectra. The E2-term of this spectral sequence consists of the right derived functors of product in the category of E*E-comodules, and the spectral sequence always converges when E is the Johnson-Wilson theory E(n) and the Xα are Ln-local. We are able to prove some results about the E2-term of this spectral sequence; in particular, we show that the E(n)-homology of a product of E(n)-module spectra Xα is just the comodule product of the E(n)*Xα. This spectral sequence is relevant to the chromatic splitting conjecture.

Given a spectrum $X$, we construct a spectral sequence of $BP_{*}BP$-comodules that converges to $BP_{*}(L_{n}X)$, where $L_{n}X$ is the Bousfield localization of $X$ with respect to the Johnson–Wilson theory $E(n)_{*}$. The $E_{2}$-term of this spectral sequence consists of the derived functors of an algebraic version of $L_{n}$. We show how to calculate these derived functors, which are closely related to local cohomology of $BP_{*}$-modules with respect to the ideal $I_{n + 1}$.

We study cubical sets without degeneracies, which we call $\square$-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a $\square$-set $C$ has an infinite family of associated $\square$-sets $J^i(C)$, for $i=1,2,\ldots$, which we call James complexes. There are mock bundle projections $p_i \colon |J^i (C)| \to |C|$ (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of $\Omega (S^2)$. The algebra of these classes mimics the algebra of the cohomotopy of $\Omega (S^2)$ and the reduction to cohomology defines a sequence of natural characteristic classes for a $\square$-set. An associated map to $BO$ leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation.

Hecke operators are used to investigate part of the ${{E}_{2}}$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of $\text{Ex}{{\text{t}}^{1}}$ which combines use of classical Hecke operators and $p$-adic Hecke operators due to Serre.

Let $G$ be a finite group, $H$ a copy of its $p$-Sylow subgroup, and $K{{\left( n \right)}^{*}}\left( - \right)$ the $n$-th Morava $K$-theory at $p$. In this paper we prove that the existence of an isomorphism between $K{{(n)}^{*}}(BG)$ and $K{{(n)}^{*}}(BH)$ is a sufficient condition for $G$ to be $p$-nilpotent.

We show that , the E-homology of the Ω-spectrum for P(n), is an E* free Hopf ring for E a complex oriented theory with In sent to 0. This covers the cases and . The generators of the Hopf ring are those necessary for the stable result. The motivation for this paper is to show that P(n) satisfies all of the conditions for the machinery of unstable cohomology operations set up in [BJW95]. This can then be used to produce splittings analogous to those for BP done in [Wil75]

We use the eta invariant to show every non-simply connected spherical space form of dimension m ≥ 5 has a countable family of non bordant metrics of positive scalar curvature.