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W. Stephen Wilson

Publications and source records attributed to W. Stephen Wilson.

13 recordsLinked to original sources
arXiv2025

The connective Morava K-theory of the second mod p Eilenberg-MacLane space

We develop tools for computing the connective n-th Morava K-theory of spaces. Starting with a Universal Coefficient Theorem that computes the cohomology version from the homology version, we show that every step in the process of computing one is mirrored in the other and that this can be used to make computations. As our example, we compute the connective n-th Morava K-theory of the second mod p Eilenberg-MacLane space.

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arXiv2024

The cohomology of the connective spectra for {K}-theory revisited

The stable mod 2 cohomologies of the spectra for connective real and complex K-theories are well known and easy to work with. However, the known bases are in terms of the anti-automorphism of Milnor basis elements. We offer simple bases in terms of admissible sequences of Steenrod operations that come from the Adem relations. In particular, the basis for the complex case is that you don't use any Steenrod operations in degree one or $2^n+1$, $n > 0$.

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arXiv2023

The connective K-theory of the Eilenberg-MacLane space K(Z/p,2)

We compute ku^*(K(Z/p,2)) and ku_*(K(Z/p,2)), the connective KU-cohomology and connective KU-homology groups of the mod-p Eilenberg-MacLane space K(Z/p,2), 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.

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arXiv2021

Stiefel-Whitney classes and immersions of orientable and Spin manifolds

We determine a nice simple formula for the largest Euclidean space for which there is an orientable n-manifold with a nonimmersion detected by Stiefel-Whitney classes. For Spin manifolds, we prove the analogue of the upper bound and establish the complete answer for n<24 and n=33,34. Results similar to many of these were obtained some 50 years ago, but in a much less tractable form. The sharp results for Spin manifolds require detailed calculations of ko-homology groups of mod-2 Eilenberg-MacLane spaces.

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arXiv2019

The Omega spectrum for Pengelley's BoP

We compute the homology of the spaces in the Omega spectrum for $BoP$. There is no torsion in $H_*(\underline{BoP}_{\; i})$ for $i \ge 2$, and things are only slightly more complicated for $i < 2$. We find the complete homotopy type of $\underline{BoP}_{\; i}$ for $i \le 6$ and conjecture the homotopy type for $i > 6$. This completes the computation of all $H_*(\underline{MSU}_{\;*})$.

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arXiv2018

The ER(2)-cohomology of X^nCP^\infty and BU(n)

We continue the development of the computability of the second real Johnson-Wilson theory. As ER(2) is not complex orientable, this gives some difficulty even with basic spaces. In this paper we compute the second real Johnson-Wilson theory for products of infinite complex projective spaces and for the classifying spaces for the unitary groups.

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arXiv2017

Landweber flat real pairs and ER(n)-cohomology

We take advantage of the internal algebraic structure of the Bockstein spectral sequence converging to ER(n)^*(pt) to prove that for spaces Z that are part of Landweber flat real pairs with respect to E(n), the cohomology ring ER(n)^*(Z) can be obtained from E(n)^*(Z) by base change. In particular, our results allow us to compute the Real Johnson-Wilson cohomology of the Eilenberg-MacLane spaces Z = K(Z, 2m+1), K(Z/2^q, 2m), K(Z/2, m) for all natural numbers $m$ and $q$, as well as connective covers of BO: BO, BSO, BSpin, and BO<8> (the last for n<3 only).

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arXiv2017

Multiplicative structure on Real Johnson-Wilson theory

We prove that the Real Johnson-Wilson theories ER(n) are homotopy associative and commutative ring spectra up to phantom maps. We further show that ER(n) represents an associatively and commutatively multiplicative cohomology theory on the category of (possibly non-compact) spaces.

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arXiv2017

The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}P^n$

The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}P^n$ are computed along with the Atiyah-Hirzebruch spectral sequence for $ER(2)^*(\mathbb{C}P^\infty)$. This, along with other papers in this series, gives us the $ER(2)$-cohomology of all Eilenberg-MacLane spaces. Since $ER(2)$ is $TMF_0(3)$ after a suitable completion, these computations also take care of that theory.

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arXiv2014

The Morava K-theory of BO(q) and MO(q)

We give an easy proof that the Morava K-theories for BO(q) and MO(q) are in even degrees. Although this is a known result, it had followed from a difficult proof that BP^*(BO(q)) was Landweber flat. Landweber flatness follows from the even Morava K-theory. We go further and compute an explicit description of K(n)_*(BO(q)) and K(n)_*(MO(q)) and reconcile it with the purely algebraic construct from Landweber flatness.

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arXiv2014

The ER(n)-cohomology of BO(q), and real Johnson-Wilson orientations for vector bundles

Using the Bockstein spectral sequence developed previously by the authors, we compute the ring ER(n)^*(BO(q)) explicitly. We then use this calculation to show that the ring spectrum MO[2^{n+1}] is ER(n)-orientable (but not ER(n+1)-orientable), where MO[2^{n+1}] is defined as the Thom spectrum for the self map of BO given by multiplication by 2^{n+1}.

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arXiv2011

Biequivariant Maps on Spheres and Topological Complexity of Lens Spaces

Weighted cup-length calculations in singular cohomology led Farber and Grant in 2008 to general lower bounds for the topological complexity of lens spaces. We replace singular cohomology by K-theory, and weighted cup-length arguments by considerations with biequivariant maps on spheres to improve on Farber-Grant's bounds by arbitrarily large amounts. Our calculations are based on the identification of key elements conjectured to generate the annihilator ideal of the toral bottom class in the ku-homology of the classifying space of a rank-2 abelian 2-group.

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