On Rigidity of the Ring Spectra Pms(P) and Ko

On rigidity of the ring spectra PmS(p) and ko

Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at

vorgelegt von Katja Hutschenreuter aus Berlin

Bonn, Mai 2012 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨atder Rheinischen Friedrich-Wilhelms-Universit¨atBonn

Diese Arbeit wurde gef¨ordertdurch ein Stipendium des DFG-Graduiertenkollegs “1150 Homotopie und Kohomologie”.

Erstgutachter: Prof. Dr. Stefan Schwede Zweitgutachter: Prof. Dr. Hans-Werner Henn Tag der Promotion: 03.09.2012 Erscheinungsjahr: 2012 On rigidity of the ring spectra PmS(p) and ko

Katja Hutschenreuter We study the rigidity question for modules over certain ring spectra. A stable model category is rigid if its homotopy category determines the Quillen equivalence type of the model category. Amongst others, we prove that the model category Mod-S is rigid if S is the mth Postnikov section of a p-localized sphere spectrum for a prime p and for a suﬃciently large integer m. Moreover, we prove that the underlying spectrum of the ring spectrum ko is determined by the ring π∗(ko) and certain Toda brackets.

Contents

Introduction 3

1 Prerequisites 7 1.1 Notation and conventions ...... 7 1.2 Model categories and rigidity ...... 8 1.2.1 Model categories and Quillen adjunctions ...... 8 1.2.2 Triangulated categories ...... 9 1.2.3 Rigidity of model categories and ring spectra ...... 12 1.3 Postnikov sections of spectra and ring spectra ...... 14

2 On rigidity of some Postnikov sections PmS(p) 17 2.1 General statements about rigidity of PmS(p) for all primes p ...... 18 2.2 The ring spectra Pm(S(2)) are rigid for all m ≥ 0 ...... 21 2 2.3 The ring spectra Pm(S(p)) are rigid for odd primes p and m ≥ p q − 1 ...... 23 2.3.1 Coherent M-modules ...... 24 2 2.3.2 The morphism ι is a π<2p−2-isomorphism for m ≥ p (2p − 2) − 1 ...... 26

3 Towards rigidity of the real connective K-theory ring spectrum 32 3.1 Cell-approximation of ko, P4ko and P8ko ...... 32 3.1.1 Gluing ring spectra cells to a ring spectrum ...... 33 3.1.2 Approximation of P4ko: Killing the element ν ∈ π3(S)...... 36 3.1.3 Approximation of P8ko and P9ko ...... 43 3.2 Cohomology of R with Z/2 -coeﬃcients ...... 47 ∧ ∧ 3.3 The ring spectra ko2 and R2 are stably equivalent as spectra ...... 56

Introduction

One main goal of algebraic topology is to understand and classify spaces up to homotopy equivalence. Depending on the context there are different definitions of homotopy, as for example homotopy between two maps of topological spaces or between two maps of simplicial sets. Moreover, the notion of homotopy does not only occur in topology but also in algebra, for example as a chain homotopy between two maps of chain complexes. In order to generalize and axiomatize these different definitions of homotopy, Quillen developed model categories [Qu]. An important part of the structure of a model category is a certain class of morphisms, the weak equivalences. Using the model structure one can define a notion of homotopy between two maps in a model category such that all weak equivalences between so-called bifibrant objects are homotopy equivalences. There are model structures on the category of topological spaces, the category of simplicial sets and the category of unbounded chain complexes of modules over a ring R, whose definitions of homotopy coincide with the classical ones on certain full subcategories. For example, the category of topological spaces can be endowed with a model structure whose weak equivalences are the morphisms that induce isomorphisms on all homotopy groups. The resulting new definition of homotopy coincides with the classical definition on all maps between cell complexes. The structure of a model category M ensures that it is possible t