Existence

Postnikov systems exist on connected CW complexes,^[1]: 354 and there is a weak homotopy-equivalence between ${\displaystyle X}$ and its inverse limit, so

${\displaystyle X\simeq \varprojlim {}X_{n}}$,

showing that ${\displaystyle X}$ is a CW approximation of its inverse limit. They can be constructed on a CW complex by iteratively killing off homotopy groups. If we have a map ${\displaystyle f\colon S^{n}\to X}$ representing a homotopy class ${\displaystyle [f]\in \pi _{n}(X)}$, we can take the pushout along the boundary map ${\displaystyle S^{n}\to e_{n+1}}$, killing off the homotopy class. For ${\displaystyle X_{m}}$ this process can be repeated for all ${\displaystyle n>m}$, giving a space which has vanishing homotopy groups ${\displaystyle \pi _{n}(X_{m})}$. Using the fact that ${\displaystyle X_{n-1}}$can be constructed from ${\displaystyle X_{n}}$ by killing off all homotopy maps ${\displaystyle S^{n}\to X_{n}}$, we obtain a map ${\displaystyle X_{n}\to X_{n-1}}$.

Main property

One of the main properties of the Postnikov tower, which makes it so powerful to study while computing cohomology, is the fact the spaces ${\displaystyle X_{n}}$ are homotopic to a CW complex ${\displaystyle {\mathfrak {X}}_{n}}$ which differs from ${\displaystyle X}$ only by cells of dimension ${\displaystyle \geq n+2}$.

Homotopy classification of fibrations

The sequence of fibrations ${\displaystyle p_{n}:X_{n}\to X_{n-1}}$^[2] have homotopically defined invariants, meaning the homotopy classes of maps ${\displaystyle p_{n}}$, give a well defined homotopy type ${\displaystyle [X]\in \operatorname {Ob} (hTop)}$. The homotopy class of ${\displaystyle p_{n}}$ comes from looking at the homotopy class of the classifying map for the fiber ${\displaystyle K(\pi _{n}(X),n)}$. The associated classifying map is

${\displaystyle X_{n-1}\to B(K(\pi _{n}(X),n))\simeq K(\pi _{n}(X),n+1)}$,

hence the homotopy class ${\displaystyle [p_{n}]}$ is classified by a homotopy class

${\displaystyle [p_{n}]\in [X_{n-1},K(\pi _{n}(X),n+1)]\cong H^{n+1}(X_{n-1},\pi _{n}(X))}$

called the n-th Postnikov invariant of ${\displaystyle X}$, since the homotopy classes of maps to Eilenberg-Maclane spaces gives cohomology with coefficients in the associated abelian group.

Postnikov tower of a K(G,n)

One of the conceptually simplest cases of a Postnikov tower is that of the Eilenberg–Maclane space ${\displaystyle K(G,n)}$. This gives a tower with

${\displaystyle {\begin{matrix}X_{i}\simeq *&{\text{for }}i<n\\X_{i}\simeq K(G,n)&{\text{for }}i\geq n\end{matrix}}}$

Postnikov tower of S²

The Postnikov tower for the sphere ${\displaystyle S^{2}}$ is a special case whose first few terms can be understood explicitly. Since we have the first few homotopy groups from the simply connectedness of ${\displaystyle S^{2}}$, degree theory of spheres, and the Hopf fibration, giving ${\displaystyle \pi _{k}(S^{2})\simeq \pi _{k}(S^{3})}$ for ${\displaystyle k\geq 3}$, hence

${\displaystyle {\begin{matrix}\pi _{1}(S^{2})=&0\\\pi _{2}(S^{2})=&\mathbb {Z} \\\pi _{3}(S^{2})=&\mathbb {Z} \\\pi _{4}(S^{2})=&\mathbb {Z} /2.\end{matrix}}}$

Then, ${\displaystyle X_{2}=S_{2}^{2}=K(\mathbb {Z} ,2)}$, and ${\displaystyle X_{3}}$ comes from a pullback sequence

${\displaystyle {\begin{matrix}X_{3}&\to &*\\\downarrow &&\downarrow \\X_{2}&\to &K(\mathbb {Z} ,4),\end{matrix}}}$

which is an element in

${\displaystyle [p_{3}]\in [K(\mathbb {Z} ,2),K(\mathbb {Z} ,4)]\cong H^{4}(\mathbb {CP} ^{\infty })=\mathbb {Z} }$.

If this was trivial it would imply ${\displaystyle X_{3}\simeq K(\mathbb {Z} ,2)\times K(\mathbb {Z} ,3)}$. But, this is not the case! In fact, this is responsible for why strict infinity groupoids don't model homotopy types.^[3] Computing this invariant requires more work, but can be explicitly found.^[4] This is the quadratic form ${\displaystyle x\mapsto x^{2}}$ on ${\displaystyle \mathbb {Z} \to \mathbb {Z} }$ coming from the Hopf fibration ${\displaystyle S^{3}\to S^{2}}$. Note that each element in ${\displaystyle H^{4}(\mathbb {CP} ^{\infty })}$ gives a different homotopy 3-type.

Definition

For a spectrum ${\displaystyle E}$ a postnikov tower of ${\displaystyle E}$ is a diagram in the homotopy category of spectra, ${\displaystyle {\text{Ho}}({\textbf {Spectra}})}$, given by

${\displaystyle \cdots \to E_{(2)}\xrightarrow {p_{2}} E_{(1)}\xrightarrow {p_{1}} E_{(0)}}$,

with maps

${\displaystyle \tau _{n}\colon E\to E_{(n)}}$

commuting with the ${\displaystyle p_{n}}$ maps. Then, this tower is a Postnikov tower if the following two conditions are satisfied:

1. ${\displaystyle \pi _{i}^{\mathbb {S} }\left(E_{(n)}\right)=0}$ for ${\displaystyle i>n}$,
2. ${\displaystyle \left(\tau _{n}\right)_{*}\colon \pi _{i}^{\mathbb {S} }(E)\to \pi _{i}^{\mathbb {S} }\left(E_{(n)}\right)}$ is an isomorphism for ${\displaystyle i\leq n}$,

where ${\displaystyle \pi _{i}^{\mathbb {S} }}$ are stable homotopy groups of a spectrum. It turns out every spectrum has a Postnikov tower and this tower can be constructed using a similar kind of inductive procedure as the one given above.

Implications

Notice ${\displaystyle X_{1}\to X}$ is the universal cover of ${\displaystyle X}$ since it is a covering space with a simply connected cover. Furthermore, each ${\displaystyle X_{n}\to X}$ is the universal ${\displaystyle n}$-connected cover of ${\displaystyle X}$.

Construction

The spaces ${\displaystyle X_{n}}$ in the Whitehead tower are constructed inductively. If we construct a ${\displaystyle K\left(\pi _{n+1}(X),n+1\right)}$ by killing off the higher homotopy groups in ${\displaystyle X_{n}}$,^[7] we get an embedding ${\displaystyle X_{n}\to K(\pi _{n+1}(X),n+1)}$. If we let

${\displaystyle X_{n+1}=\left\{f\colon I\to K\left(\pi _{n+1}(X),n+1\right):f(0)=p{\text{ and }}f(1)\in X_{n}\right\}}$

for some fixed basepoint ${\displaystyle p}$, then the induced map ${\displaystyle X_{n+1}\to X_{n}}$ is a fiber bundle with fiber homeomorphic to

${\displaystyle \Omega K\left(\pi _{n+1}(X),n+1\right)\simeq K\left(\pi _{n+1}(X),n\right)}$,

and so we have a Serre fibration

${\displaystyle K\left(\pi _{n+1}(X),n\right)\to X_{n}\to X_{n-1}}$.

Using the long exact sequence in homotopy theory, we have that ${\displaystyle \pi _{i}(X_{n})=\pi _{i}\left(X_{n-1}\right)}$ for ${\displaystyle i\geq n+1}$, ${\displaystyle \pi _{i}(X_{n})=\pi _{i}(X_{n-1})=0}$ for ${\displaystyle i<n-1}$, and finally, there is an exact sequence

${\displaystyle 0\to \pi _{n+1}\left(X_{n+1})\to \pi _{n+1}(X_{n}\right)\mathrel {\overset {\partial }{\rightarrow }} \pi _{n}K\left(\pi _{n+1}(X),n\right)\to \pi _{n}\left(X_{n+1}\right)\to 0}$,

where if the middle morphism is an isomorphism, the other two groups are zero. This can be checked by looking at the inclusion ${\displaystyle X_{n}\to K(\pi _{n+1}(X),n+1)}$ and noting that the Eilenberg–Maclane space has a cellular decomposition

${\displaystyle X_{n-1}\cup \{{\text{cells of dimension}}\geq n+2\}}$; thus,

${\displaystyle \pi _{n+1}\left(X_{n}\right)\cong \pi _{n+1}\left(K\left(\pi _{n+1}(X),n+1\right)\right)\cong \pi _{n}\left(K\left(\pi _{n+1}(X),n\right)\right)}$,

giving the desired result.

As a homotopy fiber

Another way to view the components in the Whitehead tower is as a homotopy fiber. If we take

${\displaystyle {\text{Hofiber}}(\phi _{n}:X\to X_{n})}$

from the Postnikov tower, we get a space ${\displaystyle X^{n}}$ which has

${\displaystyle \pi _{k}(X^{n})={\begin{cases}\pi _{k}(X)&k>n\\0&k\leq n\end{cases}}}$