Spectrum (topology)

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different categories of spectra, but they all determine the same homotopy category, known as the stable homotopy category.

The definition of a spectrum

There are many variations of the definition: in general, a "spectrum" is any sequence $X_{n}$ of pointed topological spaces or pointed simplicial sets together with the structure maps $S^{1}\wedge X_{n}\to X_{{n+1}}$ .

The treatment here is due to Adams (1974): a spectrum (or CW-spectrum) is a sequence $E:=\{E_{n}\}_{{n\in {\mathbb {N}}}}$ of CW-complexes together with inclusions $\Sigma E_{n}\to E_{{n+1}}$ of the suspension $\Sigma E_{n}$ as a subcomplex of $E_{{n+1}}$ .

For other definitions, see symmetric spectrum and simplicial spectrum.

Examples

Consider singular cohomology $H^{n}(X;A)$ with coefficients in an abelian group A. For a CW complex X, the group $H^{n}(X;A)$ can be identified with the set of homotopy classes of maps from X to K(A,n), the Eilenberg–MacLane space with homotopy concentrated in degree n. Then the corresponding spectrum HA has nth space K(A,n); it is called the Eilenberg–MacLane spectrum.

As a second important example, consider topological K-theory. At least for X compact, $K^{0}(X)$ is defined to be the Grothendieck group of the monoid of complex vector bundles on X. Also, $K^{1}(X)$ is the group corresponding to vector bundles on the suspension of X. Topological K-theory is a generalized cohomology theory, so it gives a spectrum. The zeroth space is ${\mathbb {Z}}\times BU$ while the first space is $U$ . Here $U$ is the infinite unitary group and $BU$ is its classifying space. By Bott periodicity we get $K^{{2n}}(X)\cong K^{0}(X)$ and $K^{{2n+1}}(X)\cong K^{1}(X)$ for all n, so all the spaces in the topological K-theory spectrum are given by either ${\mathbb {Z}}\times BU$ or $U$ . There is a corresponding construction using real vector bundles instead of complex vector bundles, which gives an 8-periodic spectrum.

For many more examples, see the list of cohomology theories.

A spectrum may be constructed out of a space. The suspension spectrum of a space X is a spectrum $X_{n}=S^{n}\wedge X$ (the structure maps are the identity.) For example, the suspension spectrum of the 0-sphere is called the sphere spectrum and is denoted by $\mathbb {S}$ .
An Ω-spectrum is a spectrum such that the adjoint of the structure map ( $X_{n}\to \Omega X_{{n+1}}$ ) is a weak equivalence. The K-theory spectrum of a ring is an example of an Ω-spectrum.
A ring spectrum is a spectrum X such that the diagrams that describe ring axioms in terms of smash products commute "up to homotopy" ( $S^{0}\to X$ corresponds to the identity.) For example, the spectrum of topological K-theory is a ring spectrum. A module spectrum may be defined analogously.

Invariants

The homotopy group of a spectrum $X_{n}$ is given by $\pi _{k}(X)=\operatorname {colim}_{{n}}\pi _{{n+k}}(X_{n})$ . Thus, for example, $\pi _{k}({\mathbb {S}})$ , $\mathbb {S}$ sphere spectrum, is the kth stable homotopy group of spheres. A spectrum is said to be connective if its $\pi _{k}$ are zero for negative k.

Functions, maps, and homotopies of spectra

There are three natural categories whose objects are spectra, whose morphisms are the functions, or maps, or homotopy classes defined below.

A function between two spectra E and F is a sequence of maps from E_n to F_n that commute with the maps ΣE_n → E_n+1 and ΣF_n → F_n+1.

Given a spectrum $E_{n}$ , a subspectrum $F_{n}$ is a sequence of subcomplexes that is also a spectrum. As each i-cell in $E_{j}$ suspends to an (i + 1)-cell in $E_{{j+1}}$ , a cofinal subspectrum is a subspectrum for which each cell of the parent spectrum is eventually contained in the subspectrum after a finite number of suspensions. Spectra can then be turned into a category by defining a map of spectra $f:E\to F$ to be a function from a cofinal subspectrum $G$ of $E$ to $F$ , where two such functions represent the same map if they coincide on some cofinal subspectrum. Intuitively such a map of spectra does not need to be everywhere defined, just eventually become defined, and two maps that coincide on a cofinal subspectrum are said to be equivalent. This gives the category of spectra (and maps), which is a major tool. There is a natural embedding of the category of pointed CW complexes into this category: it takes $Y$ to the suspension spectrum in which the nth complex is $\Sigma ^{n}Y$ .

The smash product of a spectrum $E$ and a pointed complex $X$ is a spectrum given by $(E\wedge X)_{n}=E_{n}\wedge X$ (associativity of the smash product yields immediately that this is indeed a spectrum). A homotopy of maps between spectra corresponds to a map $(E\wedge I^{+})\to F$ , where $I^{+}$ is the disjoint union $[0,1]\sqcup \{*\}$ with $*$ taken to be the basepoint.

The stable homotopy category, or homotopy category of (CW) spectra is defined to be the category whose objects are spectra and whose morphisms are homotopy classes of maps between spectra. Many other definitions of spectrum, some appearing very different, lead to equivalent stable homotopy categories.

Finally, we can define the suspension of a spectrum by $(\Sigma E)_{n}=E_{{n+1}}$ . This translation suspension is invertible, as we can desuspend too, by setting $(\Sigma ^{{-1}}E)_{n}=E_{{n-1}}$ .

The triangulated homotopy category of spectra

The stable homotopy category is additive: maps can be added by using a variant of the track addition used to define homotopy groups. Thus homotopy classes from one spectrum to another form an abelian group. Furthermore the stable homotopy category is triangulated (Vogt (1970)), the shift being given by suspension and the distinguished triangles by the mapping cone sequences of spectra

X\rightarrow Y\rightarrow Y\cup CX\rightarrow (Y\cup CX)\cup CY\cong \Sigma X

Smash products of spectra

The smash product of spectra extends the smash product of CW complexes. It makes the stable homotopy category into a monoidal category; in other words it behaves like the (derived) tensor product of abelian groups. A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy. Some more recent definitions of spectra, such as symmetric spectra, eliminate this problem, and give a symmetric monoidal structure at the level of maps, before passing to homotopy classes.

The smash product is compatible with the triangulated category structure. In particular the smash product of a distinguished triangle with a spectrum is a distinguished triangle.

Generalized homology and cohomology of spectra

We can define the (stable) homotopy groups of a spectrum to be those given by

\displaystyle \pi_n E = [\Sigma^n \mathbb{S}, E]

where $\mathbb {S}$ is the sphere spectrum and $[X,Y]$ is the set of homotopy classes of maps from $X$ to $Y$ . We define the generalized homology theory of a spectrum E by

E_n X = \pi_n (E \wedge X) = [\Sigma^n \mathbb{S}, E \wedge X]

and define its generalized cohomology theory by

\displaystyle E^{n}X=[\Sigma ^{{-n}}X,E].

Here $X$ can be a spectrum or (by using its suspension spectrum) a space.

History

A version of the concept of a spectrum was introduced in the 1958 doctoral dissertation of Elon Lages Lima. His advisor Edwin Spanier wrote further on the subject in 1959. Spectra were adopted by Michael Atiyah and George W. Whitehead in their work on generalized homology theories in the early 1960s. The 1964 doctoral thesis of J. Michael Boardman gave a workable definition of a category of spectra and of maps (not just homotopy classes) between them, as useful in stable homotopy theory as the category of CW complexes is in the unstable case. (This is essentially the category described above, and it is still used for many purposes: for other accounts, see Adams (1974) or Vogt (1970).) Important further theoretical advances have however been made since 1990, improving vastly the formal properties of spectra. Consequently, much recent literature uses modified definitions of spectrum: see Mandell et al. (2001) for a unified treatment of these new approaches.

References

Adams, J. F.(1974), "Stable homotopy and generalised homology". University of Chicago Press
Atiyah, M. F.(1961), "Bordism and cobordism", Proc. Camb. Phil. Soc. 57: 200–208
Elmendorf, A. D.; Kříž, I.; Mandell, M. A.; May, J. Peter (1995), "Modern foundations for stable homotopy theory", in James., I. M., Handbook of algebraic topology (PDF), Amsterdam: North-Holland, pp. 213–253, ISBN 978-0-444-81779-2, MR 1361891, doi:10.1016/B978-044481779-2/50007-9
Lima, Elon L. (1959), "The Spanier–Whitehead duality in new homotopy categories", Summa Brasil. Math., 4: 91–148, MR 0116332
Lages Lima, Elon (1960), "Stable Postnikov invariants and their duals", Summa Brasil. Math., 4: 193–251
Mandell, M. A.; May, J. P.; Schwede, S.; Shipley, B. (2001), "Model categories of diagram spectra", Proc. London Math. Soc. (3), 82 (2): 441–512, doi:10.1112/S0024611501012692

Vogt, Rainer (1970), Boardman's stable homotopy category, Lecture Notes Series, No. 21, Matematisk Institut, Aarhus Universitet, Aarhus, MR 0275431
Whitehead, George W. (1962), "Generalized homology theories", Trans. Amer. Math. Soc., 102 (2): 227–283, doi:10.1090/S0002-9947-1962-0137117-6

External links

Are spectra really the same as cohomology theories?

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