# nLab Introduction to Stable Homotopy Theory

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This entry is a detailed introduction to stable homotopy theory, hence to the stable homotopy category and to its key computational tool, the Adams spectral sequence. To that end we introduce the modern tools, such as model categories and highly structured ring spectra. In the accompanying seminar we consider applications to cobordism theory and complex oriented cohomology such as to converge in the end to a glimpse of the modern picture of chromatic homotopy theory.

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Lecture notes. $\;\;\;\;\;\;\;$ (web version requires Firefox browser – free download)

Prelude – Classical homotopy theory (pdf 111 pages)

Part 1 – Stable homotopy theory

$\,\,$ Part 1.1 – Sequential Spectra (pdf, 79 pages)

$\,\,$ Part 1.2 – Structured Spectra (pdf, 75 pages)

Interlude – Spectral sequences (pdf, 15 pages)

Part 2 – Adams spectral sequences (pdf, 53 pages)

Examples and Applications – Cobordism and Complex Oriented Cohomology (pdf, 76 pages)

total file (pdf, 418 pages)

Background – Introduction to Homological algebra (pdf, 83 pages)

Background – Introduction to Topology (pdf, 122 pages)

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# Contents

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My initial inclination was to call this book The Music of the Spheres, but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. (Ravenel 86, preface)

## Survey

We are concerned with the theory of spectra in the sense of algebraic topology: the proper generalization of abelian groups to homotopy theory.

### 1) Stable homotopy theory

A group in homotopy theory is equivalently a loop space under concatenation of loops (“∞-group”). A double loop space is a group with some commutativity structure (“Eckmann-Hilton argument”), a triple loop space has more commutativity structure, and so forth. A spectrum is where this progression of looping and delooping stabilizes (an “$\infty$-abelian group”). Therefore one speaks of stable homotopy theory:

$Spaces \;\; \underoverset{(linearization)}{stabilization}{\mapsto} \;\; Spectra \,.$

Most of linear algebra and algebraic geometry passes along as abelian groups are generalized to spectra and turns into something remarkably rich, called brave new algebra, higher algebra and spectral geometry. In particular the analog of the theory of (commutative) rings and their modules exists, given by (commutative) ring spectra (E-∞ rings, A-∞ rings) and module spectra (∞-modules).

Since spectra are considerably richer than abelian groups, stable homotopy is much concerned with “fracturing” stable homotopy types into more tractable components:

To that end, notice that from the point of view of arithmetic geometry, an abelian group $A$ is equivalently a quasicoherent sheaf over Spec(Z).

$AbelianGroups \simeq QCoh(Spec(\mathbb{Z})) \,.$

This point of view generalizes to homotopy theory and turns out to be very fruitful there. The analog of the integers $\mathbb{Z}$ is the sphere spectrum $\mathbb{S}$, and this is naturally the initial commutative ring spectrum (“E-∞ ring”), just as $\mathbb{Z}$ is the initial commutative ring. The formal dual Spec(S) of $\mathbb{S}$ is hence the terminal space in E-∞ arithmetic geometry (“spectral geometry”) and spectra are equivalently the quasicoherent ∞-stacks over $Spec(\mathbb{S})$

$Spectra \simeq QCoh(Spec(\mathbb{S})) \,.$

Therefore the study of spectra “fractures” into the various localizations and formal completions of $Spec(\mathbb{S})$. Since this is like the white light of $Spec(\mathbb{S})$ decomposing into various wavelengths, one speaks of chromatic homotopy theory.

In particular, an E-∞ ring $E$ is dually a morphism of $E_\infty$-algebraic spaces $Spec(E) \longrightarrow Spec(\mathbb{S})$ and under good conditions the 1-image of this map is the formal dual of the localization $L_E \mathbb{S}$ at $E$:

$Spec(E) \stackrel{epi_1}{\longrightarrow} Spec(L_E \mathbb{S}) \stackrel{mono_1}{\longrightarrow} Spec(\mathbb{S}) \,.$

This means that $Spec(E) \longrightarrow Spec(L_E \mathbb{S})$ is a cover and that hence $E$-local spectra are equivalently quasicoherent ∞-stacks on $Spec(E)$ equipped with descent data: dually they are ∞-modules over $E$ equipped with comodule structure over the Hopf algebroid (Sweedler coring) $E \otimes_{\mathbb{S}} E$.

The computation of homotopy groups of spectra that make use of their decomposition this way into $E$-∞-modules equipped with descent data is the $E$-Adams spectral sequence, a central tool of the theory.

### S) Complex oriented cohomology

For this reason special importance is carried by those E-∞ rings such that $Spec(E) \to Spec(\mathbb{S})$ is already a covering, in a suitable sense, for these the $E$-∞-modules equipped with descent data give an equivalent, but in general more tractable, incarnation of the stable homotopy theory of spectra.

Curiously, this way a good bit of differential topologycobordism theory – arises within stable homotopy theory: the archetypical $Spec(E)$ which covers $Spec(\mathbb{S})$ in a suitable sense is $E =$ MU, the Thom spectrum representing complex cobordism cohomology.

An commutative ring spectrum $E$ over $MU$, hence a $Spec(E)\to Spec(MU)$ is now a multiplicativecomplex oriented cohomology theory”.

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## Prelude) Classical homotopy theory

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This section is at: Introduction to Stable homotopy theory – P

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## Part 1) Stable homotopy theory

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This section is at Introduction to Stable homotopy theory – 1

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## Interlude) Spectral sequences

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This section is at Introduction to Stable homotopy theory – I

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## Part 2) Adams spectral sequences

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This section is at Introduction to Stable homotopy theory – 2

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## Seminar) Complex oriented cohomology

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This section is at Introduction to Stable homotopy theory – S

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## References

For Prelude) Classical homotopy theory a concise and self-contained re-write of the proof (Quillen 67) of the classical model structure on topological spaces is in

For general model category theory a decent concise account is in

For the restriction to the convenient category of compactly generated topological spaces good sources are

• Gaunce Lewis, Compactly generated spaces (pdf), appendix A of The Stable Category and Generalized Thom Spectra PhD thesis Chicago, 1978

• Neil Strickland, The category of CGWH spaces, 2009 (pdf)

For section 1) Stable homotopy theory we follow the modern picture of the stable homotopy category for which an enjoyable survey may be found in

The classical account in (Adams 74, part III sections 2, 4-7) is still a good read, but ignore the “Adams category”-construction of the stable homotopy category in sections III.2 and III.3. What we actually do follows

For the discussion of ring spectra we pass to symmetric spectra and orthogonal spectra. A compendium on the former is in

For Interlude: Spectral sequences a discussion streamlined for our purposes is in (Rognes 12, section 2).

In 2) Adams spectral sequence for the general theory we follow

For the special case of the classical Adams spectral sequence we follow (Kochman 96, chapter V).

For the Seminar on Complex oriented cohomology an excellent textbook to hold on to is

Specifically for S.1) Generalized cohomology a neat account is in:

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 12 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

For S.2) Cobordism theory an efficient collection of the highlights is in

• Cary Malkiewich, Unoriented cobordism and $M O$, 2011 (pdf)

except that it omits proof of the Leray-Hirsch theorem/Serre spectral sequence and that of the Thom isomorphism, but see the references there and see (Kochman 96, Aguilar-Gitler-Prieto 02, section 11.7) for details.

For S.3) Complex oriented cohomology besides (Kochman 96, chapter 4) have a look at Adams 74, part II and

(These overlap, pick the one that seems more inviting on first reading.)

The two originals

are still an excellent source. For further reading on homotopy theory and stable homotopy theory a useful collection is

The modern chromatic picture originates around

a useful survey is in

a wealth of details is in

and new foundations have been laid in