nLab stable homotopy theory

Idea

Stable homotopy theory is homotopy theory in the case that the operations of looping and delooping are equivalences.

As homotopy theory is the study of homotopy types, so stable homotopy theory is the study of stable homotopy types. As homotopy theory in generality is (∞,1)-category theory (or maybe (∞,1)-topos theory), so stable homotopy theory in generality is the theory of stable (∞,1)-categories.

More specifically, if one thinks of classical homotopy theory as the study of (just) the (∞,1)-category $L_{whe}$Top $\simeq$ ∞Grpd of topological spaces modulo weak homotopy equivalence (∞-groupoids), or rather of its homotopy category $Ho(Top)$, then stable homotopy theory is the study of the corresponding stabilization: To every suitable (∞,1)-category is associated its corresponding stable (∞,1)-category of spectrum objects. For $L_{whe}$Top this is the stable (∞,1)-category of spectra, $Sp(L_{whe}Top)$. Stable homotopy theory is the study of $Sp(Top)$, or rather of its homotopy category, the stable homotopy category $Ho(Sp(L_{whe}Top))$.

For detailed introduction to the stable homotopy theory of spectra see at Introduction to Stable homotopy theory.

By definition a stable homotopy type is one on which suspension and hence looping and delooping act as an equivalence. Historically people considered in plain homotopy theory statements that became true after sufficiently many suspensions, hence once the process of taking suspensions “stabilizes”. Whence the name.

Applications

Higher algebra

The study of monoid object in a monoidal (infinity,1)-category in a stable (∞,1)-category is the homotopy-theoretic version of commutative algebra, hence higher algebra and higher linear algebra.

A tool of central importance in stable homotopy theory and its application to higher algebra is the symmetric monoidal smash product of spectra which allows us to describe A-∞ rings and E-∞ rings as ordinary monoid objects in a model category that presents $Sp(Top)$. (“brave new algebra”).

Related concepts

• homotopy theory, homotopy type theory

• stable homotopy category, (infinity,1)-category of spectra

• Spanier-Whitehead duality, Anderson duality

  • S-theory

• stable (infinity,1)-category

• parameterized homotopy theory, parameterized stable homotopy theory

• rational stable homotopy theory

• cohesive stable homotopy theory

• noncommutative stable homotopy theory

• chromatic homotopy theory

• K(n)-local stable homotopy theory

• generalized homology theory, generalized cohomology theory

• stable homotopy homology theory

When the spaces and spectra in question carry an infinity-action of a group $G$ the theory refines to

• equivariant stable homotopy theory, global equivariant stable homotopy theory.

History

Stable homotopy theory began around 1937 with the Freudenthal suspension theorem. $[$…$]$ Stable phenomena had of course appeared earlier, at least implicitly: reduced homology and cohomology are examples of functors that are invariant under suspension without limitation on dimension.

Stable homotopy theory emerged as a distinct branch of algebraic topology with Adams’ introduction of his eponymous spectral sequence and his spectacular conceptual use of the notion of stable phenomena in his solution to the Hopf invariant one problem.

Its centrality was reinforced by two related developments that occurred at very nearly the same time, in the late 1950’s. One was the introduction of generalized homology and cohomology theories and especially K-theory, by Atiyah and Hirzebruch. The other was the work of Thom which showed how to reduce the problem of classifying manifolds up to cobordism to a problem, more importantly, a solvable problem, in stable homotopy theory $[$ Thom spectrum $]$.

The reduction of geometric phenomena to solvable problems in stable homotopy theory has remained an important mathematical theme, the most recent major success being Stolz’s use of Spin cobordism to study the classication of manifolds with positive scalar curvature.

In an entirely different direction, the early 1970’s saw Quillen’s introduction of higher algebraic K-theory and the recognition by Segal and others that it could be viewed as a construction in stable homotopy theory. With algebraic K-theory as an intermediary, there has been a growing volume of work that relates algebraic geometry to stable homotopy theory. With Waldhausen’s introduction of the algebraic K-theory of spaces in the late 1970’s, stable homotopy became a bridge between algebraic K-theory and the study of diffeomorphisms of manifolds.

Within algebraic topology, the study of stable homotopy theory has been and remains the focus of much of the best work in the subject.

(Elmendorf-Kriz-May 95, p. 2)

References

The original direct definitions of the stable homotopy category (for precursors see at Spanier-Whitehead category) is due to

• Michael Boardman, Stable homotopy theory, mimeographed notes, University of Warwick, 1965 onward

Early accounts:

• Rainer Vogt, Boardman’s stable homotopy category, lectures, spring 1969

• J. M. Cohen, Stable Homotopy, Springer Lecture Notes in Math., No. 165, Springer-Verlag, Berlin, 1970.

• Dieter Puppe, On the stable homotopy category, Topology and its application (1973) (pdf)

• Frank Adams, Part III, section 2 of Stable homotopy and generalised homology, 1974

• Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

A fun scan of the (pre-)history of the stable homotopy category:

• Peter May, The hare and the hedgehog, speech at Michael Boardman‘s birthday meeting (txt)

Quick surveys:

• Cary Malkiewich, The stable homotopy category, 2014 (pdf, pdf)

• Dylan Wilson, Introduction to stable homotopy theory (pdf, pdf)

• Aaron Mazel-Gee, An introduction to spectra (pdf)

Lecture notes:

• Michael Hopkins (notes by Akhil Mathew), Spectra and stable homotopy theory, 2012 (pdf, pdf)

• Urs Schreiber, Introduction to Stable homotopy theory, Bonn 2016

• David Barnes, Constanze Roitzheim, Foundations of Stable Homotopy Theory, Cambridge University Press 2020 (doi:10.1017/9781108636575)

• Denis Nardin, Introduction to stable homotopy theory, Regensburg 2021 (webpage, pdf, video)

See also:

• Glossary for stable and chromatic honotopy theory (pdf)

Models with symmetric monoidal smash product of spectra:

for S-modules:

• Anthony Elmendorf, Igor Kriz, Peter May, Modern foundations for stable homotopy theory, in Ioan Mackenzie James, Handbook of Algebraic Topology, Amsterdam: North-Holland, (1995) pp. 213–253, (pdf)

for symmetric spectra

• Stefan Schwede, Symmetric spectra (pdf)

for orthogonal spectra:

• Stefan Schwede, Global homotopy theory (take $\mathcal{F} = \{1\}$, on p. 4, to be the trivial collection of groups, in order to specialize from global equivariant stable homotopy theory to plain stable homotopy theory).

A survey of formalisms used in stable homotopy theory – tools to present the triangulated homotopy category of a stable (infinity,1)-category – is in

• Neil Strickland, Axiomatic stable homotopy - a survey (arXiv:math.AT/0307143)

• Mark Hovey, John Palmieri, Neil Strickland, Axiomatic stable homotopy theory, Memoirs Amer. Math. Soc. 610 (1997) [ISBN:978-1-4704-0195-5, pdf]

An account in terms of (∞,1)-category theory is in section 1 of

• Jacob Lurie, Higher Algebra

Brief indications of open questions and future directions (as of 2013) of algebraic topology and stable homotopy theory are in

• Tyler Lawson, The future, Talbot lectures 2013 (pdf)