Spectrum of spaces

A representing object for a generalized cohomology theory. The notion was introduced in [1] (cf. also Generalized cohomology theories).

A spectrum $ \mathbf M $ of spaces is defined as a sequence of topological (as a rule cellular) spaces $ \{ M _ {n} \} _ {n= - \infty } ^ \infty $ together with mappings $ S _ {n} : \Sigma M _ {n} \rightarrow M _ {n+} 1 $, where $ \Sigma $ denotes suspension. Spectra of spaces form a category; a morphism of a spectrum $ \mathbf M $ into a spectrum $ \mathbf N $ is, roughly speaking, the "cofinal part" of some function $ f: \mathbf M \rightarrow \mathbf N $ given by a family of mappings $ f _ {n} : M _ {n} \rightarrow N _ {n} $ with $ t _ {n} \circ \Sigma f _ {n} = f _ {n+} 1 \circ s _ {n} $( $ t _ {n} $ is to $ \mathbf N $ as $ s _ {n} $ is to $ \mathbf M $). The concepts of homotopic morphisms and homotopy equivalent spectra of spaces may be introduced, and one may construct the homotopy category of spectra [2]. Postnikov systems (cf. Postnikov system) of spectra of spaces have also been introduced.

The suspension $ \Sigma \mathbf M $ over a spectrum of spaces $ \mathbf M $ is defined as the spectrum of spaces $ \Sigma \mathbf M = \{ N _ {n} \} = \{ \Sigma M _ {n} \} $. Let $ ( \Sigma ^ {-} 1 \mathbf M ) _ {n} = M _ {n-} 1 $. Then $ \Sigma $ and $ \Sigma ^ {-} 1 $ are homotopically mutually inverse functors, so that in the category of spectra of spaces (in contrast to the category of spaces), the suspension functor is invertible, and this makes it convenient to work in the former. In general, in the category of spectra of spaces, all arguments connected with stability (e.g. the construction of the Adams spectral sequence) assume a natural form.

Examples of spectra of spaces.

1) For any space $ X $ one can define a spectrum of spaces $ \mathbf X = \{ M _ {n} \} $, where $ M _ {n} = * $ when $ n< 0 $ and $ M _ {n} = \Sigma ^ {n} X $ when $ n \geq 0 $, and $ s _ {n} $ is the natural identification $ \Sigma ( \Sigma ^ {n} X) \rightarrow \Sigma ^ {n+} 1 X $. So, for $ X = S ^ {0} $ one obtains the spectrum of spheres $ \{ S ^ {n} \} $.

2) The spectrum of the Eilenberg–MacLane spaces $ H( \pi ) $( or $ \mathbf{EM} ( \pi ) $), where $ \pi $ is an Abelian group. The homotopy equivalence $ \omega _ {n} : K( \pi , n) \rightarrow \Omega K ( \pi , n+ 1) $, where $ K( \pi , n) $ is an Eilenberg–MacLane space and $ \Omega X $ is the loop space over $ X $, gives an adjoint mapping $ s _ {n} : \Sigma K ( \pi , n) \rightarrow K ( \pi , n+ 1) $, so that one obtains the spectrum of spaces $ \{ K ( \pi , n), s _ {n} \} $. This spectrum of spaces represents the ordinary cohomology theory with coefficients in $ \pi $.

3) Let $ X $ be a space such that $ \Omega ^ {d} X \simeq X $ for some $ d $. For $ n= ad+ b $, $ 0\leq b \leq d $, $ a \in \mathbf Z $, let $ M _ {n} = \Omega ^ {d-} b X $. There arises a sequence $ \{ M _ {n} \} $ of the form $ \{ \dots, X, \Omega ^ {d-} 1 X , \Omega ^ {d-} 2 X \dots \Omega ^ {1} X , X \simeq \Omega ^ {d} X ,\dots \} $. As in example 2), the homotopy equivalence $ \omega : M _ {n} \rightarrow \Omega M _ {n+} 1 $ gives a mapping $ s _ {n} : \Sigma M _ {n} \rightarrow M _ {n+} 1 $, so that one obtains a spectrum of spaces. E.g., for the classifying space $ \mathop{\rm BU} = \lim\limits \mathop{\rm BU} _ {n} $, where the $ \textrm{ U } _ {n} $ are unitary groups, $ \Omega ^ {2} ( \mathop{\rm BU} \times \mathbf Z ) \simeq \mathop{\rm BU} \times \mathbf Z $( Bott's periodicity theorem), and one obtains the spectrum of spaces $ \{ \dots, U, \mathop{\rm BU} \times \mathbf Z , U, \mathop{\rm BU} \times \mathbf Z ,\dots \} $ representing complex $ K $- theory. An analogous result holds for real $ K $- theory ( $ \Omega ^ {8} ( \mathop{\rm BO} \times \mathbf Z _ {2} ) \simeq \mathop{\rm BO} \times \mathbf Z _ {2} $).

4) Various Thom spectra (cf. Thom spectrum), representing cobordism theories.

Given two spectra of spaces $ \mathbf M $ and $ \mathbf N $, one can define their reduced product $ \mathbf M \wedge \mathbf N $( an analogue of the usual reduced product of spaces). A multiplication on $ \mathbf M $ is defined as a morphism $ \mathbf M \wedge \mathbf M \rightarrow \mathbf M $ which is associative (in the appropriate sense). A spectrum of spaces equipped with a multiplication is called a ring spectrum, or multiplicative spectrum, and the cohomology theory represented by it is multiplicative. Attempts to overcome the difficulties connected with the "poor associativity" of the above multiplication have led to a revision of the foundations of the theory of spectra of spaces. Namely, the concept of a coordinate-free spectrum of spaces is introduced as a family of spaces $ \{ M _ {V} \} $( and corresponding mappings) indexed by linear subspaces $ V $ of $ \mathbf R ^ \infty = \lim\limits _ {n \rightarrow \infty } \mathbf R ^ {n} $. The category of coordinate-free spectra of spaces is isomorphic to the category of ordinary spectra of spaces, but the pairing $ \wedge $ in it is easier to control, and so it plays an important role in the examination of subtle geometric questions connected with the higher structures of spectra of spaces, with orientations in cohomology theory, and elsewhere.

References

[1]	E.L. Lima, "The Spanier–Whitehead duality in new homotopy categories" Summa Brasiliens. Math. , 4 (1959) pp. 91–148
[2]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975)
[3]	J.F. Adams, "Infinite loop spaces" , Princeton Univ. Press (1978)
[4]	M. Flato, "Deformation view of physical theories" Czechoslovak J. Phys. , B32 (1982) pp. 472–475
[5]	J.P. May, "-ring spaces and -ring spectra" , Lect. notes in math. , 577 , Springer (1977)

Comments

For the cohomology and homology theory defined by a spectrum see Generalized cohomology theories; for the adjointness of suspension and loop, and hence the canonical mapping $ X \rightarrow \Omega \Sigma X $, $ x \mapsto \omega _ {x} $, $ \omega _ {x} ( t)=( x, t) $, see Suspension.

A mapping of spectra $ f: \mathbf M \rightarrow \mathbf N $ is defined by individual continuous mappings $ f _ {n} : M _ {n} \rightarrow N _ {n} $( or $ M _ {n} \rightarrow N _ {n-} r $ for a mapping of spectra of degree $ r $). Applying the mapping-cone construction and mapping cylinder construction to these individual mappings defines the mapping cone of a mapping of spectra and the mapping cylinder of a mapping of spectra.

A spectrum $ \mathbf M $ is an $ \Omega $- spectrum if the mappings $ s _ {n} ^ \prime : M _ {n} \rightarrow \Omega M _ {n+} 1 $( adjoint to $ s _ {n} : \Sigma M _ {n} \rightarrow M _ {n+} 1 $) are weak homotopy equivalences.

Often, a CW-spectrum is defined as a sequence $ \{ M _ {n} \} $ of CW-complexes $ M _ {n} $ such that $ \Sigma M _ {n} $ is (or, is homotopic to) a subcomplex of $ M _ {n+} 1 $.

References

[a1]	B. Eckmann, "Homotopy and cohomology theory" , Proc. Internat. Congress Mathematicians (Stockholm, 1962) , Almqvist & Wiksells (1963) pp. 59–75
[a2]	E.H. Brown, "Cohomology theories" Ann. of Math. , 75 (1962) pp. 467–484
[a3]	G.W. Whitehead, "Generalized homology theories" Trans. Amer. Math. Soc. , 102 (1962) pp. 227–238