Spectrum (homotopy theory)

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, all of which give the same homotopy category.

Suppose we start with a generalized cohomology theory E. This is a sequence of contravariant functors $E^n$ from topological spaces to abelian groups, one for each integer n, which satisfy all of the Eilenberg-Steenrod axioms except for the dimension axiom. By the Brown representability theorem, $E^n(X)$ is given by $[X,E\_n]$, the set of homotopy classes of maps from X to $E\_n$, for some space $E\_n$. The isomorphism $E^n(X)\; cong\; E^\{n+1\}(Sigma\; X)$, where $Sigma\; X$ is the suspension of X, gives a map $Sigma\; E\_n\; o\; E\_\{n+1\}$. This collection of spaces $E\_n$ together with connecting maps $Sigma\; E\_n\; o\; E\_\{n+1\}$ is a spectrum. In most (but not all) constructions of spectra the adjoint maps $E\_n\; o\; Omega\; E\_\{n+1\}$ are required to be weak equivalences or even homeomorphisms.

We can also construct homology and cohomology theories given a particular spectrum. We restrict attention to spectra whose spaces are pointed CW-complexes. Given a spectrum $E\_n$, a subspectrum $F\_n$ is a sequence of subcomplexes that is also a spectrum. Noting that each i-cell in $E\_j$ becomes 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 $f:\; E\; o\; F$ to be a cofinal subspectrum $G$ of $E$ and a sequence of pointed maps $f\_n:\; G\_n\; o\; F\_n$ such that $Sf\_n\; =\; f\_\{n+1|G\_n\}$ (i.e. the obvious square commutes). 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. 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 spectra corresponds to a map $(E\; wedge\; I^+)\; o\; F$, where $I^+$ is the disjoint union $[0,\; 1]\; sqcup\; \{*\}$ with * taken to be the basepoint. Finally, we can define the suspension of a spectrum as $(Sigma\; E)\_n\; =\; E\_\{n+1\}$.

Given all this information, we can first note that there is an inverse to the suspension functor given by $(Sigma^\{-1\}\; E)\_\{n+1\}\; =\; E\_n$. We can define the homotopy groups of a spectrum to be those given by $pi\_n\; E\; =\; [Sigma^n\; S,\; E]$, where $S$ is the spectrum of spheres and $[X,\; Y]$ is the set of homotopy classes of maps from $X$ to $Y$. Using some facts about what are commonly called "cofiber sequences of spectra" we arrive at the definitions $E\_n\; X\; =\; pi\_n\; (E\; wedge\; X)\; =\; [Sigma^n\; S,\; E\; wedge\; X]$ and $E^n\; X\; =\; [Sigma^\{-n\}\; S\; wedge\; X,\; E]$ for the homology and cohomology theories respectively associated to the spectrum $E$. It is worth noting that $S\_n\; X\; =\; pi\_n(S\; wedge\; X)$ corresponds to the nth stable homotopy group of $X$.

Examples

Consider singular cohomology $H^n(X;A)$ with coefficients in an abelian group A. By Brown representability $H^n(X;A)$ is 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 n'th 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 zero'th space is $mathbf\{Z\}\; imes\; BU$ 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 $mathbf\{Z\}\; imes\; BU$ 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.

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. There was development of the topic by J. Michael Boardman, amongst others. The above setting came together during the mid-1960s, and is still used for many purposes: 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 (and highly technical) definitions of spectrum: see Mandell "et al." (2001) for a unified treatment of these new approaches.

References

* J. F. Adams (1974). "Stable homotopy and generalised homology". University of Chicago Press.

* citation
first = M. A.|last= Mandell|first2=J. P.|last2= May,|first3= S. |last3=Schwede |first4=B. |last4=Shipley
year = 2001
title = Model categories of diagram spectra
journal = Proc. London Math. Soc. (3)
volume = 82
pages = 441-512
doi=10.1112/S0024611501012692

* R. Vogt (1970). "Boardman's stable homotopy category". Lecture note series No. 21, Matematisk Institut, Aarhus University.