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Definition 1.1. [Swi02] A spectrum \(E=\{(E_n,e_n, \varphi _n)\}_{n\geq 0}\) contains the following information:
A morphism of spectra \(f=\{f_n\}_{n\geq 0}: E=\{(E_n,e_n, \varphi _n)\}_{n\geq 0}\to F=\{(F_n,f_n, \psi _n)\}_{n\geq 0}\) contains pointed map \(f_n: (E_n,e_n)\to (F_n,f_n)\) such that
Here \(\Omega \) is the loop space functor. Note the definition is equivalent to having a sequence of pointed map \(\varphi _n': \Sigma E_n\to E_{n+1}\) due to the adjunction \(\Sigma \vdash \Omega : \Spc _*\to \Spc _*\), where \(\Sigma \) is the suspension functor, and \(\Spc _*\) is the homotopy category of pointed spaces (we let \(\Spc \) denote the unpointed case)
Definition 1.2. [Swi02] Given a spectrum \(E:=\{(E_n, \varphi _n)\}\) and a pointed CW complex \(X\), we define the smash product \(E\wedge X\) as \(\{(E_n\wedge X, \overline {\varphi _n \wedge id _X})\}_{n \geq 0}\), where \(\overline {\varphi _n \wedge id _X}:E_n\wedge X\to \Omega (E_{n+1}\wedge X)\) is the adjoint of \(\varphi _n' \wedge id_X: \Sigma (E_n\wedge X)=\Sigma E_n\wedge X\to E_{n+1}\wedge X\).
Definition 1.3. [Swi02] Given two spectra morphisms \(f,g :E\to F\), a homotopy between \(f,g\) is a map \(H: E\wedge I_+\to F\), such that \(H|_{0}=f\) and \(H|_1=g\), and we call \(f\) homotopic to \(g\). A homotopy equivalence is a pair of map \(f:E\to F: g\), such that \(f\circ g\) is homotopic to \(id_F\) and \(g\circ f\) is homotopic to \(id_E\).
We denote \([E,F]\) the homotopy class of morphisms of spectra from \(E\) to \(F\).
Definition 1.4. [Rud98] A \(\Omega \)-spectrum is a spectrum \(E=\{(E_n, \varphi _n )\}_{n \geq 0}\) such that the adjunction map \(\varphi _n': E_n\to \Omega E_{n+1}\) is a weak homotopy equivalence.
Notice, for a \(\Omega \)-spectrum \(E\), the stable homotopy group \(\pi _kE\) is just the ordinary homotopy group of \(\pi _k \Omega ^\infty E\), as \(\pi _{n+k}E_n\to \pi _{n+1+k}E_{n+1}\) are isomorphisms for all \(n\).
Definition 1.5. [Swi02] A morphism of spectra \(f:E\to F\) is a weak homotopy equivalence if \(f\) induces isomorphism \(\pi _kf: \pi _k E\to \pi _k F\) for all \(k\in \Z \) (not just nonnegative ones).
Theorem 1.6. [Rud98] Every spectrum is weakly equivalent to a \(\Omega \)-spectrum.
Proof. Define functor \(Q: \Spc _*\to \Spc _*\) as \begin{align*} Q(X)=\colim _n \Omega ^n\Sigma ^n X \end{align*}
and for spectrum \(E\), define \(Q(E)=\{(Q(E_n), Q(\varphi _n))\}_{n\geq 0}\).
There is natural transformation \(id_{\Spc _*}\implies Q\) from the simply inclusion, which determines a map \(E\to Q(E)\). Notice \(\pi _{n+k}E_n\to \pi _{n+k}Q(E_n)\) is an isomorphism for large enough \(k\). □
If one is familiar with the language of model categories, we can develop a pretty nice model category of spectra with fibrant-cofibrant objects being the \(\Omega \)-spectra. Of course, everything still works even if we refrain from going into the technical details. Let us denote \(\Sp \) the homotopy category of spectra, so the objects are \(\Omega \)-spectra and morphisms are homotopy class.
The category \(\Sp \) is pointed, so it has a zero object \(0\).
Definition 1.7. Define functor \(\Sigma ^\infty : \Spc _*\to \Sp \) such that \(\Sigma ^\infty X=\{(X_n,id)\}\), where \(X_0=X\), \(X_n=\Sigma X_{n-1}\), and \(id: \Sigma X_n\to X_{n+1}\) is the identity. We call this the suspension spectrum functor and we refer to \(\Sigma ^\infty X\) as the suspension spectrum of \(X\).
Define functor \(\Omega ^\infty : \Sp \to \Spc _*\) such that \(\Omega ^\infty E=E_0\). We call this the infinite loop space functor and refer to \(\Omega ^\infty E\) as the infinite loop space of \(Y\).
The name suspension spectrum functor makes sense, and the infinite loop space is because \(\Omega ^\infty \Sigma ^\infty X\) is precisely \(Q(X)\) in the homotopy category.
Given a spectrum \(E=\{(E_n, \varphi _n)\}_{n\geq 0}\), for every \(k\in \Z \), there is a sequence of group homomorphisms \begin{align*} ...\to \pi _{n+k}E_n\to \pi _{n+k}\Omega \Sigma E_{n}\cong \pi _{n+1+k}\Sigma E_{n}\xrightarrow {\pi _{n+1+k}\varphi _n} \pi _{n+1+k} E_{n+1}\to ... \end{align*}
Where if \(k<0\), \(\pi _k\) is understand to be zero.
Definition 1.8. Given a spectrum \(E\), the stable homotopy group \(\pi _k E\) is defined as \begin{align*} \pi _kE:=\colim _{n} \pi _{n+k}E_n \end{align*}
The motivation for these definitions comes from a theorem.
Theorem 1.9. (Freudenthal’s Suspension Theorem) Let \(X\) be a pointed \(n\)-connected CW complex, then for any \(k\), the map \begin{align*} \pi _kX\to \pi _{k}\Omega \Sigma X=\pi _{k+1}\Sigma X \end{align*}
induced from the adjunction unit is an isomorphism if \(k\leq 2n\) and epimorphism if \(k=2n+1\).
Suppose we have any pointed CW complex \(X\), \(\Sigma ^{n}X\) will be at least \(n-1\)-connected. Therefore, the map \begin{align*} \pi _{n+k}\Sigma ^{n}X\to \pi _{(n+1)+k}\Sigma ^{n+1}X \end{align*}
is an isomorphism if \(n+k\leq 2(n-1)\), or equivalently, \(k\leq n-2\). Moreover, \begin{align*} \pi _{(n+1)+k}\Sigma ^{n+1}X\to \pi _{(n+2)+k}\Sigma ^{n+2}X \end{align*}
is an isomorphism if \(k\leq n-1\).
In other words, as we go down the sequence, the homotopy groups will stabilize. Therefore, the stable homotopy group \(\pi _k\Sigma ^\infty X\) is just \(\pi _k X_m\) for large enough \(m\).
Definition 1.10. We define functor \(\Omega : \Sp \to \Sp \) such that \(\Omega E=\{(E_{n+1}, \varphi _{n+1})\}_{n+1\geq 0}\), so we shift everything down. There is functor \(\Sigma :\Sp \to \Sp \) such that \(\Sigma E=\{(E_{n-1}, \varphi _{n-1})\}_{n-1\geq 0}\), where we define \(E_{-1}=\Omega E_0\).
Notice, there is \(\Omega \Sigma \simeq \Sigma \Omega \simeq id\).
We denote \(S:=\Sigma ^\infty S^0\). Since we have \(\Sigma \vdash \Omega : \Spc _*\to \Spc _*\), for any spectra \(E\), there is \(\pi _kE\cong [\Sigma ^kS, E]\).
Theorem 1.11. There exists adjunction \(\Sigma ^\infty \vdash \Omega ^\infty : \Spc _*\to \Sp \), i.e. \([\Sigma ^\infty X, E]\cong [X, \Omega ^\infty E]_*\)
Proof. Given \(f:\Sigma ^\infty X\to E\), there is
Notice, that \(f_0\) determines \(f_n\) uniquely. □
The details of definitions below require the knowledge of the model structure on the category of spectra, which is not necessary for understanding the note.
Definition 1.12. Let \(F: I\to \Sp \) be a homotopy coherent diagram of spectra, we denote \(\holim _IF\) the homotopy limit of \(F\) in the model category of spectra, and \(\hocolim _I F\) the homotopy colimit in the model category of spectra.
We specifically look at the homotopy cofiber sequence. For every \(f: E\to F\), we define \(Cf:=\{ (F_n\cup _{f_n}CE_n, \varphi _n)\}\), where \(\varphi _n\) is induced from the structure map \(\Sigma F_n\to F_{n+1}\) and \(\Sigma (CE_n)=C(\Sigma E_n)\to CE_{n+1}\).
Proposition 1.13. [Swi02] For every morphism \(f: E\to F\), there is a homotopy cofiber sequence \begin{align*} E\xrightarrow {f}F\to Cf\to \Sigma E \end{align*}
Proof. Homotopy cofiber in a simplicial model category (for instance \(\Sp \)) is just ordinary pushout with the point in the diagram replacement with the cone, so \(Cf\) determines the homotopy cofiber of \(f\).
We have homotopy pasting law, so we just need to show the homotopy pushout of \(E\) is \(\Sigma E\) (which we guess is true from the notation chosen). We have that the homotopy pushout will be \(E'\) where \(E'_0=*\) and \(E'_n=\Sigma E_{n-1}\). There is map \(f: E' \to \Sigma E\) induced from the structure map \(\Sigma E_{n-1}\to E_n\), which we know induces isomorphism on the stable homotopy groups. □
We have a similar result regarding homotopy fiber. Given map \(f: E\to F\), let \(Pf\) be the homotopy fiber of \(f\). There is then homotopy fiber sequence \[ \Omega F\to Pf\to E\xrightarrow {f}F \]
Proposition 1.14. A sequence \(E\to F\to G\) is a homotopy cofiber sequence if and only if it is a homotopy fiber sequence.
In other words, we can extend every cofiber(equivalently fiber) sequence \(E\to F\to G\) into a long exact sequence \[ ...\to \Omega F\to \Omega G\to E\to F\to G \to \Sigma E\to \Sigma F\to ... \] where any two adjacent map determines a cofiber/fiber sequence.
Since every spectrum is a looping of its suspension, so the homotopy class of spectra morphisms are abelian groups (in fact, the suspension and loop space functor, with the additive property of the homotopy category of spectra determines a triangulated category)
Proposition 1.15. Suppose \(E\to F\to G\) is a cofiber sequence, then for any spectrum \(X\) \begin{align*} [X, E]\to [X, F]\to [X, G] \\ [G, X]\to [F, X]\to [E, X] \end{align*}
are exact sequences of abelian groups.
Theorem 1.16. [Rud98] There exists a symmetric monoidal structure \(\wedge \) on \(\Sp \) such that
Definition 1.17. [Swi02] A ring spectrum is a tuple \((E, \psi , e)\) with \(\psi : E\wedge E\to E\) and \(e: S\to E\), such that
and
commutes (up to homotopy). Moreover, if
commutes (up to homotopy), then we call it a commutative ring spectrum.
Definition 1.18. Given a ring spectra \((E,\psi , e)\), a left (resp. right) \(E\)-module is a tuple \((M, \phi )\), where \(\phi : E\wedge M\to M\) (resp. \(\phi : M\wedge E\to M\)) such that
the left diagram commute up to homotopy (resp. right diagram commute up to homotopy).