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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 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 Xn{displaystyle X_{n}} of pointed topological spaces or pointed simplicial sets together with the structure maps S1∧Xn→Xn+1{displaystyle S^{1}wedge X_{n}to X_{n+1}}.

The treatment here is due to Frank Adams (1974): a spectrum (or CW-spectrum) is a sequence E:={En}n∈N{displaystyle E:={E_{n}}_{nin mathbb {N} }} of CW complexes together with inclusions ΣEn→En+1{displaystyle Sigma E_{n}to E_{n+1}} of the suspension ΣEn{displaystyle Sigma E_{n}} as a subcomplex of En+1{displaystyle E_{n+1}}.

For other definitions, see symmetric spectrum and simplicial spectrum.

Examples

Consider singular cohomology Hn(X;A){displaystyle H^{n}(X;A)} with coefficients in an abelian group A. For a CW complex X, the group Hn(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, K0(X){displaystyle K^{0}(X)} is defined to be the Grothendieck group of the monoid of complex vector bundles on X. Also, K1(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×BU{displaystyle mathbb {Z} times BU} while the first space is U{displaystyle U}. Here U{displaystyle U} is the infinite unitary group and BU{displaystyle BU} is its classifying space. By Bott periodicity we get K2n(X)≅K0(X){displaystyle K^{2n}(X)cong K^{0}(X)} and K2n+1(X)≅K1(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×BU{displaystyle mathbb {Z} times 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.

• A spectrum may be constructed out of a space. The suspension spectrum of a space X is a spectrum Xn=Sn∧X{displaystyle 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 S{displaystyle mathbb {S} }.


• An Ω-spectrum is a spectrum such that the adjoint of the structure map (Xn→ΩXn+1{displaystyle 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" (S0→X{displaystyle 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 Xn{displaystyle X_{n}} is given by πk(X)=colimn⁡πn+k(Xn){displaystyle pi _{k}(X)=operatorname {colim} _{n}pi _{n+k}(X_{n})}. Thus, for example, πk(S){displaystyle pi _{k}(mathbb {S} )}, S{displaystyle mathbb {S} } sphere spectrum, is the kth stable homotopy group of spheres. A spectrum is said to be connective if its πk{displaystyle 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 En{displaystyle E_{n}}, a subspectrum Fn{displaystyle F_{n}} is a sequence of subcomplexes that is also a spectrum. As each i-cell in Ej{displaystyle E_{j}} suspends to an (i + 1)-cell in Ej+1{displaystyle 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→F{displaystyle f:Eto F} to be a function from a cofinal subspectrum G{displaystyle G} of E{displaystyle E} to F{displaystyle 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{displaystyle Y} to the suspension spectrum in which the nth complex is ΣnY{displaystyle Sigma ^{n}Y}.

The smash product of a spectrum E{displaystyle E} and a pointed complex X{displaystyle X} is a spectrum given by (E∧X)n=En∧X{displaystyle (Ewedge 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∧I+)→F{displaystyle (Ewedge I^{+})to F}, where I+{displaystyle I^{+}} is the disjoint union [0,1]⊔{∗}{displaystyle [0,1]sqcup {*}} with ∗{displaystyle *} 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 (ΣE)n=En+1{displaystyle (Sigma E)_{n}=E_{n+1}}. This translation suspension is invertible, as we can desuspend too, by setting (Σ−1E)n=En−1{displaystyle (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→Y→Y∪CX→(Y∪CX)∪CY≅ΣX{displaystyle Xrightarrow Yrightarrow Ycup CXrightarrow (Ycup CX)cup CYcong 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

πnE=[ΣnS,E]{displaystyle displaystyle pi _{n}E=[Sigma ^{n}mathbb {S} ,E]},

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

EnX=πn(E∧X)=[ΣnS,E∧X]{displaystyle E_{n}X=pi _{n}(Ewedge X)=[Sigma ^{n}mathbb {S} ,Ewedge X]}

and define its generalized cohomology theory by

EnX=[Σ−nX,E].{displaystyle displaystyle E^{n}X=[Sigma ^{-n}X,E].}

Here X{displaystyle 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 Rainer 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 Michael Mandell et al. (2001) for a unified treatment of these new approaches.

See also

• Ring spectrum


• Symmetric spectrum


• G-spectrum


• Mapping spectrum


• Suspension (topology)

References

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


• 
  Atiyah, Michael F. (1961). "Bordism and cobordism". Proceedings of the Cambridge Philosophical Society. 57 (2): 200–208. doi:10.1017/s0305004100035064..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  Elmendorf, A. D.; Kříž, I.; Mandell, M. A.; May, J. Peter (1995), "Modern foundations for stable homotopy theory" (PDF), in James., I. M., Handbook of algebraic topology, Amsterdam: North-Holland, pp. 213–253, CiteSeerX 10.1.1.55.8006, doi:10.1016/B978-044481779-2/50007-9, ISBN 978-0-444-81779-2, MR 1361891
  


• 
  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, Michael A.; May, J. Peter; Schwede, Stefan; Shipley, Brooke (2001), "Model categories of diagram spectra", Proceedings of the London Mathematical Society, Series 3, 82 (2): 441–512, CiteSeerX 10.1.1.22.3815, doi:10.1112/S0024611501012692, MR 1806878
  


• 
  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", Transactions of the American Mathematical Society, 102 (2): 227–283, doi:10.1090/S0002-9947-1962-0137117-6
  

External links

• 
  "Are spectra really the same as cohomology theories?".