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Dissertation

zur Erlangung des Doktorgrades (Dr. rer. nat.)

der Fakult¨at fur¨ Mathematik

der Ruhr-Universit¨at Bochum

vorgelegt von

Dipl.-Math. Holger Reeker

Mai 2009Contents

Chapter 1. Introduction and statement of results 5

Chapter 2. Some homotopical algebra 9

1. Generalized cohomology theories and spectra 9

2. Symmetric spectra over topological spaces 11

3. Complex oriented theories and computational methods 14

4. Bousﬁeld localization of spectra 15

5. Algebraic manipulations of spectra 18

6. A resolution of the K(1)-local sphere 20

7. Thom isomorphism 21

Chapter 3. The algebraic structure of K(1)-local E ring spectra 23∞

1. Operads 23

2. Dyer-Lashof operations for K(1)-local E spectra 25∞

3. The θ-algebra structure of π K∧MU 260

4. The θ str of π K∧MSU 290

Chapter 4. Splitting oﬀ an E summand T 31∞ ζ

1. The image of MSU →MU 32∗ ∗

2. Formal group laws and Miscenkos formula 34

ˆ3. Construction of an SU-manifold with A = 1 38

4. Constr of an Artin-Schreier class 41

5. Construction of an E map T →MSU 41∞ ζ

6. Split map - direct summand argument 42

0Chapter 5. Detecting free E summands TS 45∞

1. Introduction to Adams operations in K-theory 45

∞2. Calculating operations on K CP 47∗

3. Bott’s formula and cannibalistic classes 52

4. Spherical classes in K MSU 55∗

5. Umbral calculus 57

Chapter 6. Open questions and concluding remarks 61

Bibliography 63

3CHAPTER 1

Introduction and statement of results

Oneofthehighlightsinalgebraictopologywastheinventionofgeneralizedhomologyand

cohomologytheories by Whiteheadand Brown in the 1960s. Prominent examples are real

and complex K-theories ﬁrst given by Atiyah and Hirzebruch and bordism theories with

respect to diﬀerent structure groups ﬁrst given by Thom. By Brown’s representability

theoremeverygeneralizedcohomologytheorycanberepresentedbyaspectrumandthese

spectra are the center of interest in modern algebraic topology.

Bordism theories with respect to some structure group G, e.g. G = O, SO, U, SU, Sp,

andSpin are deﬁned as follows: LetM be a smooth, closed,n-dimensional manifold and

G ={G} be a sequence of topological groups with maps G → G compatible withn n n+1

their orthogonal representations G →O(n).n

Deﬁnition 1.1. AG-structure onM is a homotopy class of liftsν˜ of the classifying map

of the stable normal bundle ν

BG

ν˜

ν

M BO

A manifold M together with a G-structure is called a G-manifold.

GForeachoftheclassicalgroupsthisgivesustheG-bordismringΩ andaThomspectrum∗

GMGwithΩ =MG =MG (pt) =π MG. FurtherwehaveahomologytheoryMG (−)∗ ∗ ∗ ∗∗

∗andacohomologytheoryMG (−). Sincewehaveinclusionmapsongrouplevelandsince

the Thom construction is functorial we get the following tower:

MSpinMSU

MU MSO

MO

On the level of homotopy one knows at least rationally that the coeﬃcient groups are

polynomial rings and one asks for a decomposition on the level of spectra. In 1966

Andersen, Brown and Peterson gave an additive 2-local splitting of MSpin

_ _ _

diMSpin ’ koh4n(J)i∨ koh4n(J)+2i∨ Σ HZ/2(2)

i∈In(J) even, 1∈/J n(J) odd, 1∈/J

with J = (i ,...,i ) a ﬁnite sequence and n(J) =i +...+i . Bordism theories are mul-1 k 1 k

tiplicative homology theories and their Thom spectra are ring spectra. Moreover they

5

/;;/////6 1. INTRODUCTION AND STATEMENT OF RESULTS

admit even richer structures called E structures, i.e. not only the coherent diagrams of∞

commutativity and associativity commute up to homotopy but there are also diagrams

of higher coherence. These E structures should be taken into account and therefore we∞

are interested in a splitting in the category of E ring spectra.∞

Unfortunately this access raises several other diﬃculties. Analysing the above addi-

tive splitting of 2-local spin bordism by Anderson, Brown and Peterson, the Eilenberg-

MacLane part HZ/2 turns out to be a diﬃcult problem. In this situation the modern

viewpoint is to apply chromatic homotopy theory and to look at the chromatic tower or

at certain monochromatic layers. In our case we consider localizations with respect to the

ﬁrst Morava K-theory K(1). At p = 2 we have

∼L =L LK(1) SZ/2 K(2)

and the Eilenberg-MacLane part disappears. This is our approximation to bordism theo-

ries. Algebraically this access oﬀers a lot of extra structure sinceπ E of aK(1)-localE0 ∞

ring spectrum E admits a θ-algebra structure.

In [Lau01] Laures gives a K(1)-local splitting of E spectra∞

∞^

0∼MSpin =T ∧ TSζ

i=1

whereT isthefreefunctorleftadjointtotheforgetfulfunctorfromE spectratospectra∞V W

0 0∼and∧ is the coproduct in the category of E spectra with TS T( S ). Such a=∞

splitting is also desireable for other bordism theories and a lot of diﬀerent techniques are

involved to get such a splitting.

InthisworkwestudyK(1)-localSU bordism. AmainresultisdetectinganE summand∞

0∼T for a nontrivial element ζ∈π L S Z=ζ −1 K(1) 2

ζ

−1 0TS S

T∗

0 0 TζTD =S

MSU

meaning that T is the resulting E spectrum when attaching a 0-cell along ζ. To thisζ ∞

3end, we construct an Artin-Schreier class b∈ KO MSU satisfying ψ b = b + 1 which0

implies that ζ = 0 in π MSU.−1

Another important result is the construction of spherical classes in K MSU. Although∗

we do not have a complete splitting, comparison with the spin bordism case shows that

0spherical classes play an important role: They correspond to free E summands TS .∞

In this work, we perform the construction of spherical classes via calculations of Adams

∞ ∞operations on K (CP ×CP ) whose module generators map to the algebra generators∗

of K BSU. Later we can use Bott’s theory of cannibalistic classes to lift the Adams∗

operations to the level of Thom spectra.

∞Since the K-homology of CP is isomorphic to the ring of numerical polynomials, we

,//#,/#/1. INTRODUCTION AND STATEMENT OF RESULTS 7

∞are able to provide an alternative calculation of the Adams operations on K CP using∗

Mahler series expansion in p-adic analysis.

Acknowledgements. First of all, I would like to thank my supervisor Prof. Dr.

Gerd Laures for introducing me to the ﬁeld ofK(1)-localE spectra with all their inter-∞

esting arithmetic. I would like to express my profound respect to Prof. Dr. Uwe Abresch

for his spontaneous willingness to act as co-referee. At the same time, I want to say

thank you to all members of the chair of topology at the Ruhr-Universitat¨ Bochum for

the good atmosphere and for numerous mathematical and non-mathematical discussions

– my special thanks go to Dr. Markus Szymik, Dr. Hanno von Bodecker, Jan M¨ollers,

Norman Schumann and Sieglinde Fernholz. I appreciate the ﬁnancial support from the

DFG within the Graduiertenkolleg 1150 “Homotopy and Cohomology”.CHAPTER 2

Some homotopical algebra

1. Generalized cohomology theories and spectra

In this section we want to recall the basic notations of generalized cohomology theories

and spectra as their representing objects. We will see the correspondence between them

and have a look at their fundamental properties. The relevant homotopy category is the

stable homotopy category.

nDeﬁnition 2.1. A generalized cohomology theory E consists of a sequence{E } ofn∈Z

contravariant homotopy functors

nE :CWPairs→AbGroups

together with natural transformations

n n+1δ :E (X)→E (X,A)

satisfying the axioms

• Excision: The projection (X,A)→X/A induces an isomorphism

n n˜E (X/A)→E (X,A)

for all pairs (X,A).

• Exactness: The long sequence of abelian groups

δn n n n+1...→E (X,A)→E (X)→E (A)→E (X,A)→...

is exact for all pairs (X,A).

• Strong additivity: For every family of spaces{X} the natural mapi i∈I

a Y

n nE ( X )→ E (X )i i

i∈I i∈I

is an isomorphism.

Proposition2.1. Every generalized cohomology theoryE enjoys the following properties:

(1) For a pointed topological space X there is a natural direct sum splitting

n n n∼ ˜E (X) =E (X)⊕E (∗).

(2) For a family{X} of pointed topological spaces the map of reduced cohomologyi i∈I

groups

_ Y

n n˜ ˜E ( X )→ E (X )i i

i∈I i∈I

is an isomorphism.

(3) For a pointed topological space X we have natural isomorphisms

∼ ∼= =n+1n n+1˜ ˜E (CX,X)E (X) E (ΣX)

δ excision

9

/o/o10 2. SOME HOMOTOPICAL ALGEBRA

(4) Mayer-Vietoris: For X = X ∪X (open covering) we have the long exact1 2

sequence

∗ ∗ ∗ ∗(i ,i ) j −j δ1 2 1 2n n n n n+1...→E (X) → E (X )⊕E (X ) → E (X ∩X )→E (X)→...1 2 1 2

(5) Milnor sequence: For a ﬁltration X = colimX we get a short exact sequencei

with the derived limit

1 n−1 n n0→ lim E (X )→E (X)→ limE (X )→ 0i i

which detects phantom maps.

These cohomology functors are representable by a sequence of spaces and with the sus-

pension isomorphism we naturally get the following deﬁnition:

Deﬁnition 2.2. A spectrum X is a sequence of pointed topological spaces X ,X ,X ,...0 1 2

together with structure maps

1σ :X ∧S →Xn n n+1

or the adjoint map σ˜ : X → ΩX respectively. If σ˜ is a weak equivalence X is calledn n+1

an Ω-spectrum.

By Brown’s representability theorem every generalized cohomology theory can be repre-

sented by an Ω-spectrum. On the other hand every spectrum deﬁnes a cohomology (and

homology) theory. It is worth mentioning that every spectrum can functorially be turned

into an Ω-spectrum. As an illustrative example, we deﬁne an Ω-spectrum for complex

K-theory.

Example 2.1. (K-theory) First of all we make use of Bott periodicity, i.e. there is a

2 ∼homotopy equivalence Ω BU =Z×BU, and we deﬁne an Ω-spectrum K by setting

(

Z×BU if n is even,

K =n

ΩBU if n is odd

with structure maps adjoint to

(

∼= 2the Bott equivalence Z×BU−→ Ω BU if n is even,

(σ˜ :K → ΩK ) =n n+1 ∼=

the identiﬁcation ΩBU−→ Ω(Z×BU) if n is odd.

This is called the complex topological K-theory spectrum. Its homotopy groups are

(

∼π (Z×BU) Z if n is even=0

π K =n

π BU = 0 if n is odd.1

Example 2.2. (KO-theory) Similarly, we obtain the real topological K-theory spectrum

KO using real Bott periodicity, i.e.

8 ∼Ω BO Z×BO.=

Its homotopy groups are given in the following table:

n mod 8 0 1 2 3 4 5 6 7

π KO Z Z/2 Z/2 0 Z 0 0 0n