Spectrum (topology)

In the mathematical sub-area of algebraic topology , spectra are used to define generalized homology theories.

definition

A spectrum is a sequence of dotted spaces with dotted continuous images ${\displaystyle E_{n}}$

${\displaystyle \sigma _{n}\colon SE_{n}\to E_{n+1}}$.

Here the reduced hanging of . ${\displaystyle SE_{n}}$ ${\displaystyle E_{n}}$

Because the reduced suspension is left adjoint to the formation of the loop space , corresponds to a continuous mapping that is unambiguous except for homotopy . A spectrum is a spectrum if the maps are homeomorphisms . ${\displaystyle \sigma _{n}}$ ${\displaystyle E_{n}\to \Omega E_{n+1}}$ ${\displaystyle \Omega }$ ${\displaystyle E_{n}\to \Omega E_{n+1}}$

Other definitions can be found in the literature. For example, the spectra defined above are called the pre-spectrum and the spectra are then called the spectrum . With these designations one can assign each presectrum through a spectrum, its spectrification . ${\displaystyle \Omega }$ ${\displaystyle E_{n}}$ ${\displaystyle \operatorname {colim} _{k\to \infty }\Omega ^{k}E_{n+k}}$

A morphism between spectra and is a family of continuous mappings with for all . ${\displaystyle (E_{n},\sigma _{n})}$ ${\displaystyle (E_{n}^{\prime },\sigma _{n}^{\prime })}$ ${\displaystyle f_{n}\colon E_{n}\to E_{n}^{\prime }}$ ${\displaystyle f_{n+1}\circ \sigma _{n}=\sigma _{n}^{\prime }\circ Sf_{n}}$ ${\displaystyle n}$

Examples

• Hanging spectra : For a topological space forms a spectrum with the canonical homeomorphisms . It is called the suspension spectrum of the room . More generally, spectra of the form are referred to as hanging spectra , with the spectrum being meant for a spectrum . ${\displaystyle X}$ ${\displaystyle E_{n}:=S^{n}X}$ ${\displaystyle \sigma _{n}\colon SE_{n}\to E_{n+1}}$ ${\displaystyle \Sigma ^{\infty }X}$ ${\displaystyle X}$ ${\displaystyle \Sigma ^{k}\Sigma ^{\infty }X}$ ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle E=(E_{n},\sigma _{n})}$ ${\displaystyle \Sigma ^{k}E}$ ${\displaystyle (E_{n+k},\sigma _{n+k})}$
• Spherical spectrum : The hanging spectrum of the -dimensional sphere is called the spherical spectrum and is denoted by. So in this case and is canonical homeomorphism. ${\displaystyle 0}$ ${\displaystyle \mathbf {S} }$ ${\displaystyle E_{n}=S^{n}}$ ${\displaystyle \sigma _{n}\colon SS^{n}\to S^{n+1}}$
• Eilenberg-MacLane spectrum : For an Abelian group , the Eilenberg-MacLane spaces form a spectrum with and the homotopy equivalence given by Whitehead's theorem . This spectrum is also referred to as. ${\displaystyle G}$ ${\displaystyle E_{n}=K(G,n)}$ ${\displaystyle \sigma _{n}\colon SK(G,n)\to K(G,n+1)}$ ${\displaystyle HG}$
• Thom spectrum : The Thom spaces of the universal vector bundles over the Graßmann manifolds form a spectrum . In this case, the structure mapping is the mapping induced by the classifying mapping of the vector bundle ${\displaystyle Th(\tau _{n})}$ ${\displaystyle \tau _{n}\to BO(n)}$ ${\displaystyle BO(n)}$ ${\displaystyle MO(n)}$ ${\displaystyle BO(n)\to BO(n+1)}$ ${\displaystyle (\tau _{n}\oplus {\underline {\mathbb {R} }})\to BO(n)}$ ${\displaystyle \sigma _{n}:SMO(n)=STh(\tau _{n})=Th(\tau _{n}\oplus {\underline {\mathbb {R} }})\to Th(\tau _{n+1})=MO(n+1).}$
• Topological K-theory spectrum : This spectrum is defined by for all , where is the ascending union of the unitary groups and their classifying space. ${\displaystyle E_{2n}=U,E_{2n+1}=\mathbb {Z} \times BU}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle U}$ ${\displaystyle BU}$
• ${\displaystyle \Omega }$-Spectra : Let be an infinite loop space, then define a -spectra. ${\displaystyle X}$ ${\displaystyle E_{n}=\Omega ^{-n}X}$ ${\displaystyle \Omega }$
• Algebraic K-theory spectrum : For a commutative ring with one , the application of the plus construction to the classifying space of , is an infinite loop space and therefore defines a spectrum. ${\displaystyle R}$ ${\displaystyle X=BGL^{+}(R)}$ ${\displaystyle GL(R)}$ ${\displaystyle \Omega }$

Homotopy groups of spectra

The kth homotopy group of a spectrum is defined by

${\displaystyle \pi _{k}(E):=\operatorname {colim} _{n\to \infty }\pi _{n+k}E_{n}}$.

The homotopy groups of a hanging spectrum are called stable homotopy groups of : ${\displaystyle E_{n}=S^{n}X}$ ${\displaystyle X}$

${\displaystyle \pi _{k}^{s}(X):=\pi _{k}(\Sigma ^{\infty }X)}$.

The following already applies to spectra . ${\displaystyle \Omega }$ ${\displaystyle \pi _{k}(E)=\pi _{k}(E_{0})}$

Examples

• The stable homotopy groups of the spheres are the homotopy groups of the sphere spectrum . ${\displaystyle \mathbf {S} }$
• The algebraic K-theory of a commutative ring with one is obtained for by definition as homotopy groups of the algebraic K-theory spectrum. ${\displaystyle K_{i}(R)}$ ${\displaystyle R}$ ${\displaystyle i\geq 1}$
• The cobordism group of unoriented -manifolds is isomorphic to the -th homotopy group of the Thom spectrum. ${\displaystyle n}$ ${\displaystyle n}$

Equivalences

The following analogue of Whitehead's Theorem applies to morphisms of spectra :

A morphism of spectra induces an isomorphism of all homotopy groups if and only if the induced morphism in the homotopy category of the spectra is an isomorphism. Such mappings are called equivalences .

Generalized theories of homology

A spectrum defines a (reduced) generalized homology theory through

${\displaystyle {\tilde {E}}_{k}(X):=\left[\Sigma ^{k}\mathbf {S} ,E\wedge X\right]}$,

where with the aid of the smash product by designated defined spectrum. ${\displaystyle E\wedge X}$ ${\displaystyle (E\wedge X)_{n}:=E_{n}\wedge X}$

In particular is . ${\displaystyle {\tilde {E}}_{k}(S^{0})=\pi _{k}(E)}$

example

${\displaystyle MO_{k}(X)}$is isomorphic to the cobordism group of singular -manifolds in . ${\displaystyle k}$ ${\displaystyle X}$

Generalized cohomology theories

Each spectrum defines a generalized (reduced) cohomology theory through ${\displaystyle E}$

${\displaystyle {\widetilde {E}}^{k}(X):=\operatorname {colim} _{n\to \infty }\left[S^{n}X,E_{n+k}\right]}$

for topological spaces , where the homotopy classes denote dotted continuous mappings. (The theory of cohomology is said to be represented by the spectrum.) ${\displaystyle X}$ ${\displaystyle \left[.,.\right]}$

The associated unreduced cohomology theory is denoted by. ${\displaystyle E^{*}(X)}$

Examples

The Eilenberg-MacLane spectrum defines the singular cohomology , the topological K-theory spectrum defines topological K-theory . ${\displaystyle K(G,n)}$ ${\displaystyle H^{*}(X;G)}$

calculation

Generalized cohomology groups of a room can often be calculated using the Atiyah-Hirzebruch spectral sequence. This is a converging spectral sequence with -term ${\displaystyle E^{*}}$ ${\displaystyle X}$ ${\displaystyle E^{*}(X)}$ ${\displaystyle E_{2}}$

${\displaystyle E_{2}^{pq}=H^{p}(X;E^{q}(X))}$,

where singular cohomology denotes a coefficient group . ${\displaystyle H^{*}(.;E^{q}(X))}$ ${\displaystyle E^{q}(X)}$

Brownian representability theorem

From Brown's representability theorem it follows that every reduced generalized cohomology theory can be represented by a spectrum. ${\displaystyle \Omega }$

Smash product

For a spectrum and a space , the spectrum is defined by and the structure mappings . ${\displaystyle E=(E_{n},\sigma _{n})}$ ${\displaystyle X}$ ${\displaystyle E\wedge X}$ ${\displaystyle (E\wedge X)_{n}:=E_{n}\wedge X}$ ${\displaystyle \Sigma _{n}\wedge id_{X}}$

There is a construction going back to Adams that assigns two spectra and a smash product , which has the following properties: ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E\wedge F}$

• The smash product is a covariant functor of both arguments.
• There are natural equivalences . ${\displaystyle \alpha \colon (E\wedge F)\wedge G\to E\wedge (F\wedge G),\tau \colon E\wedge F\to F\wedge E,l\colon \mathbf {S} \wedge E\to E,r\colon E\wedge \mathbf {S} \to E,\Sigma \colon \Sigma E\wedge F\to \Sigma (E\wedge F)}$
• There is a natural equivalence for every spectrum and every CW complex . Especially for all CW complexes . ${\displaystyle E}$ ${\displaystyle X}$ ${\displaystyle e\colon E\wedge X\to E\wedge \Sigma ^{\infty }X}$ ${\displaystyle \Sigma ^{\infty }(X\wedge Y)\simeq \Sigma ^{\infty }X\wedge \Sigma ^{\infty }Y}$ ${\displaystyle X,Y}$
• If there is an equivalence, then too . ${\displaystyle f\colon E\to F}$ ${\displaystyle f\wedge id_{G}\colon E\wedge G\to F\wedge G}$
• For a family of Spektra is an equivalence. ${\displaystyle \left\{E_{\Lambda }\right\}}$ ${\displaystyle \left\{i_{\Lambda }\wedge id_{F}\right\}\colon \vee _{\Lambda }(E_{\Lambda }\wedge F)\to (\vee _{\Lambda }(E_{\Lambda }))\wedge F}$
• If there's a fiber structure from Spektra, then so is it . ${\displaystyle A\to B\to C}$ ${\displaystyle A\wedge E\to B\wedge E\to C\wedge E}$

Ring spectra

A ring spectrum is a spectrum with a smash product and with morphisms ${\displaystyle R}$ ${\displaystyle \wedge }$

${\displaystyle \mu \colon R\wedge R\to R,\epsilon \colon \mathbf {S} \to R,}$

${\displaystyle \alpha \colon R\wedge (R\wedge R)\to (R\wedge R)\wedge R,\lambda \colon \mathbf {S} \wedge R\to R,\rho \colon R\wedge \mathbf {S} \to R}$,

which the conditions

${\displaystyle \mu \circ (\mu \wedge id)\circ \alpha =\mu \circ (id\wedge \mu )\colon R\wedge (R\wedge R)\to R}$

${\displaystyle \mu \circ (\epsilon \wedge id)=\lambda \colon \mathbf {S} \wedge R\to R,\mu \circ (id\wedge \epsilon )=\rho \colon R\wedge \mathbf {S} \to R}$

suffice.

literature

• Spanier, EH; Whitehead, JHC: A first approximation to homotopy theory. Proc. Nat. Acad. Sci. USA 39, (1953). 655-660. pdf
• Lima, Elon L .: Stable Postnikov invariants and their duals. Summa Brasil. Math. 4 1960 193-251.
• Adams, JF: Stable homotopy and generalized homology. Reprint of the 1974 original. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1995. ISBN 0-226-00524-0

Web links

• Greenlees: Spectra for commutative algebraists
• Rognes: The sphere spectrum
• Malkiewich: The stable homotopy category
• Swede: Symmetric spectra