We compute $ku^*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$ and $ku_*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$, the connective $KU$-cohomology and connective $KU$-homology groups of the mod-$p$ Eilenberg–MacLane space $K\!\left({\mathbb{Z}}_p,2\right)$, using the Adams spectral sequence. We obtain a striking interaction between $h_0$-extensions and exotic extensions. The mod-$p$ connective $KU$-cohomology groups, computed elsewhere, are needed in order to establish higher differentials and exotic extensions in the integral groups.

We prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin–Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabiliser group. As an application, we compute the zeroth limit at all primes and heights.

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.

Let $X^{n}$ be an oriented closed generalized $n$-manifold, $n\ge 5$. In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597–607), we have constructed a map $t:\mathcal {N}(X^{n}) \to H^{st}_{n} ( X^{n}; \mathbb{L}^{+})$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$-manifold. Here, $\mathcal {N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,\,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$. An important non-trivial question arose whether the map $t$ is bijective (note that this holds in the case when $X^{n}$ is a topological $n$-manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.

We observe that every self-dual ternary code determines a holomorphic $\mathcal N=1$ superconformal field theory. This provides ternary constructions of some well-known holomorphic $\mathcal N=1$ superconformal field theories (SCFTs), including Duncan’s “supermoonshine” model and the fermionic “beauty and the beast” model of Dixon, Ginsparg, and Harvey. Along the way, we clarify some issues related to orbifolds of fermionic holomorphic CFTs. We give a simple coding-theoretic description of the supersymmetric index and conjecture that for every self-dual ternary code this index is divisible by $24$ ; we are able to prove this conjecture except in the case when the code has length $12$ mod $24$ . Lastly, we discuss a conjecture of Stolz and Teichner relating $\mathcal N=1$ SCFTs with Topological Modular Forms. This conjecture implies constraints on the supersymmetric indexes of arbitrary holomorphic SCFTs, and suggests (but does not require) that there should be, for each k, a holomorphic $\mathcal N=1$ SCFT of central charge $12k$ and index $24/\gcd (k,24)$ . We give ternary code constructions of SCFTs realizing this suggestion for $k\leq 5$ .

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.

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.

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.

The $ER\left( 2 \right)$-cohomology of $B\mathbb{Z}/\left( {{2}^{q}} \right)$ and $\mathbb{C}{{\mathbb{P}}^{n}}$ are computed along with the Atiyah–Hirzebruch spectral sequence for $ER{{\left( 2 \right)}^{*}}\left( \mathbb{C}{{\mathbb{P}}^{\infty }} \right)$. This, along with other papers in this series, gives us the $ER\left( 2 \right)$-cohomology of all Eilenberg–MacLane spaces.

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.

In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the $b$–principle and in dimension 2 approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension 2 to a closed simply connected manifold is bordant to a submersion. We show that the Madsen–Weiss theorem (the standard Mumford Conjecture) fits a general setting of the $b$–principle, namely, a version of the $b$–principle for oriented colored broken submersions together with the Harer stability theorem and Miller–Morita theorem implies the Madsen–Weiss theorem.

A theorem due to Ohkawa states that the collection of Bousfield equivalence classes of spectra is a set. We extend this result to arbitrary combinatorial model categories.

B. Schuster [19] proved that the mod 2 Morava K-theory K(s)*(BG) is evenly generated for all groups G of order 32. For the four groups G of order 32 with the numbers 38, 39, 40 and 41 in the Hall-Senior list [11], the ring K(2)*(BG) has been shown to be generated as a K(2)*-module by transferred Euler classes. In this paper, we show this for arbitrary s and compute the ring structure of K(s)*(BG). Namely, we show that K(s)*(BG) is the quotient of a polynomial ring in 6 variables over K(s)*(pt) by an ideal for which we list explicit generators.

We discuss the analytic aspects of the geometric model for K-homology with coefficients in ℤ/kℤ constructed in [12]. In particular, using results of Rosenberg and Schochet, we construct a map from this geometric model to its analytic counterpart. Moreover, we show that this map is an isomorphism in the case of a finite CW-complex. The relationship between this map and the Freed-Melrose index theorem is also discussed. Many of these results are analogous to those of Baum and Douglas in the case of spinc manifolds, geometric K-homology, and Atiyah-Singer index theorem.

The actions, anomalies and quantization conditions allow the M2-brane and the M5-brane to support, in a natural way, structures beyond spin on their world-volumes. The main examples are twisted string structures. This also extends to twisted stringc structures which we introduce and relate to twisted string structures. The relation of the C-field to Chern–Simons theory suggests the use of the string cobordism category to describe the M2-brane.

Our results are of three types. First, we describe a general procedure of adjoining polynomial variables to $A_\infty$-ring spectra whose coefficient rings satisfy certain restrictions. A host of examples of such spectra is provided by killing a regular ideal in the coefficient ring of $MU$, the complex cobordism spectrum. Second, we show that the algebraic procedure of adjoining roots of unity carries over in the topological context for such spectra. Third, we use the developed technology to compute the homotopy types of spaces of strictly multiplicative maps between suitable $K(n)$-localizations of such spectra. This generalizes the famous Hopkins–Miller theorem and gives strengthened versions of various splitting theorems.

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]