Spectrum (homotopy theory)

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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 $Σ X$ is the suspension of X, gives a map $\Sigma E_n \to E_{n+1}$ . This collection of spaces $E n$ together with connecting maps $\Sigma E_n \to E_{n+1}$ is a spectrum. In most (but not all) constructions of spectra the adjoint maps $E_n \to \Omega E_{n+1}$ are required to be weak equivalences or even homeomorphisms.

[edit] 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 group completion 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 $\mathbb{Z} \times BU$ while the first space is $U$ . Here $U$ is the infinite unitary group and $B U$ 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.

[edit] 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.

[edit] References

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

Mandell, M. A.; May,, J. P.; Schwede, S. & Shipley, B. (2001), “Model categories of diagram spectra”, Proc. London Math. Soc. (3) 82: 441-512, DOI 10.1112/S0024611501012692