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

Donald M Davis

TL;DR

This work determines the connective KO-homology and KO-cohomology of the mod 2 Eilenberg-MacLane space K({\nBbb Z}_2,2) via the Adams spectral sequence, building on complex KU results and Morava K-theory inputs. It provides a detailed KO-decomposition of ko_*(K_2) into summands \mathcal{A}_k and z^i\mathcal{B}_{k,\ell}, with explicit edge structures and exotic extensions, and establishes a duality ko_*(K_2) ≅ (ko^{*+6}(K_2))^\vee. The authors derive formulaic edge descriptions, derive KO-differentials from corresponding ku-differentials, and extend the summand framework to ko^*(K_2) with A_k^* and (z^i\mathcal{B}_{k,\ell})^* components. A key application is a new obstruction result for Spin manifolds: there exists an n-dimensional Spin manifold with nonzero dual Stiefel-Whitney class \overline{w}_{n-2} iff n is a 2-power at least 8, tying KO-homology to immersion obstructions. Overall, the paper advances the structure theory of ko_*(K_2) and ko^*(K_2) and their connections to classical characteristic classes and Spin geometry.

Abstract

We compute ko_*(K(Z/2,2)) and ko^*(K(Z/2,2)), the connective KO-homology and -cohomology of the mod 2 Eilenberg-MacLane space K(Z/2,2), using the Adams spectral sequence. The work relies heavily on work done several years earlier for the (complex) ku groups by the author and W.S.Wilson. We illustrate an interesting duality relation between the ko-homology and -cohomology groups. We deduce a new result about Stiefel-Whitney classes in Spin manifolds.

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

TL;DR

This work determines the connective KO-homology and KO-cohomology of the mod 2 Eilenberg-MacLane space K({\nBbb Z}_2,2) via the Adams spectral sequence, building on complex KU results and Morava K-theory inputs. It provides a detailed KO-decomposition of ko_*(K_2) into summands \mathcal{A}_k and z^i\mathcal{B}_{k,\ell}, with explicit edge structures and exotic extensions, and establishes a duality ko_*(K_2) ≅ (ko^{*+6}(K_2))^\vee. The authors derive formulaic edge descriptions, derive KO-differentials from corresponding ku-differentials, and extend the summand framework to ko^*(K_2) with A_k^* and (z^i\mathcal{B}_{k,\ell})^* components. A key application is a new obstruction result for Spin manifolds: there exists an n-dimensional Spin manifold with nonzero dual Stiefel-Whitney class \overline{w}_{n-2} iff n is a 2-power at least 8, tying KO-homology to immersion obstructions. Overall, the paper advances the structure theory of ko_*(K_2) and ko^*(K_2) and their connections to classical characteristic classes and Spin geometry.

Abstract

We compute ko_*(K(Z/2,2)) and ko^*(K(Z/2,2)), the connective KO-homology and -cohomology of the mod 2 Eilenberg-MacLane space K(Z/2,2), using the Adams spectral sequence. The work relies heavily on work done several years earlier for the (complex) ku groups by the author and W.S.Wilson. We illustrate an interesting duality relation between the ko-homology and -cohomology groups. We deduce a new result about Stiefel-Whitney classes in Spin manifolds.

Paper Structure

This paper contains 22 sections, 24 theorems, 119 equations, 2 tables.

Table of Contents

  1. Introduction and overview
  2. Examples of $\mathcal{A}_k$
  3. One description of $\mathcal{A}_k$
  4. $z^i\mathcal{B}_{k,\ell}$
  5. Complete description through grading 42
  6. A formulaic description of edges
  7. Outline of proof
  8. Using the $ku^*$ differentials
  9. Summand approach to $ku^*(K_2)$
  10. The summands for $\mathcal{A}_k$ and $\mathcal{B}_{k,\ell}$
  11. Computation of $\mathcal{A}_4$ and $\mathcal{A}_1$
  12. Derivation of edge description
  13. The spectral sequence for $\mathcal{A}_k$, with proofs
  14. Description of spectral sequence
  15. Justification of differentials
  16. ...and 7 more sections

Key Result

Theorem 1.1

There exists an $n$-dimensional Spin manifold with the dual Stiefel-Whitney class $\overline{w}_{n-2}\ne0$ if and only if $n$ is a $2$-power $\ge8$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 3.1
  • Definition 3.5
  • Definition 3.7
  • Definition 3.8
  • Theorem 3.10
  • Definition 3.18
  • Definition 3.19
  • ...and 38 more