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. 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. There are many variations of the definition: in general, a spectrum is any sequence X n {displaystyle X_{n}} of pointed topological spaces or pointed simplicial sets together with the structure maps S 1 ∧ X n → X n + 1 {displaystyle S^{1}wedge X_{n} o X_{n+1}} . The treatment here is due to Frank Adams (1974): a spectrum (or CW-spectrum) is a sequence E := { E n } n ∈ N {displaystyle E:={E_{n}}_{nin mathbb {N} }} of CW complexes together with inclusions Σ E n → E n + 1 {displaystyle Sigma E_{n} o E_{n+1}} of the suspension Σ E n {displaystyle Sigma E_{n}} as a subcomplex of E n + 1 {displaystyle E_{n+1}} . For other definitions, see symmetric spectrum and simplicial spectrum. Consider singular cohomology H n ( X ; A ) {displaystyle H^{n}(X;A)} with coefficients in an abelian group A. For a CW complex X, the group H n ( X ; A ) {displaystyle H^{n}(X;A)} can be identified with the set of homotopy classes of maps from X to K ( A , n ) {displaystyle K(A,n)} , the Eilenberg–MacLane space with homotopy concentrated in degree n. Then the corresponding spectrum HA has nth space K ( A , n ) {displaystyle 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 ) {displaystyle K^{0}(X)} is defined to be the Grothendieck group of the monoid of complex vector bundles on X. Also, K 1 ( X ) {displaystyle 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 Z × B U {displaystyle mathbb {Z} imes BU} while the first space is U {displaystyle U} . Here U {displaystyle U} is the infinite unitary group and B U {displaystyle BU} is its classifying space. By Bott periodicity we get K 2 n ( X ) ≅ K 0 ( X ) {displaystyle K^{2n}(X)cong K^{0}(X)} and K 2 n + 1 ( X ) ≅ K 1 ( X ) {displaystyle 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 Z × B U {displaystyle mathbb {Z} imes BU} or U {displaystyle 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.