Postnikov system

In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree $k$ agree's with the truncated homotopy type of the original space $X$ . Postnikov systems were introduced by, and named after, Mikhail Postnikov.

Definition

A Postnikov system of a path-connected space $X$ is an inverse system of spaces

\cdots \to X_{n}\to \cdots \to X_{2}\to X_{1}

with a sequence of maps $\phi _{n}\colon X\to X_{n}$ compatible with the inverse system such that

(1) $\phi _{n}\colon X\to X_{n}$ induces an isomorphism $\pi _{i}(X)\to \pi _{i}(X_{n})$ for every $i\leq n$

(2) $\pi _{i}(X_{n})=0$ for $i>n$ [1]

There is an additional technical condition which some authors include as an additional axiom:

(3) Each map $X_{n}\to X_{n-1}$ is a fibration, and so the fiber $F_{n}$ is an Eilenberg–Maclane space, $K(\pi _{n}(X),n)$ .

Note that the first two conditions imply $X_{1}$ is also a $K(\pi _{1}(X),1)$ -space; more generally, if $X$ is $(n-1)$ -connected, then $X_{n}$ is a $K(\pi _{n}(X),n)$ -space and all $X_{i}$ for $i<n$ are contractible.

Existence

Postnikov systems exist on connected CW complexes [2] and there is a weak homotopy-equivalence between $X$ and $\varprojlim {}X_{n}$ showing $X$ is a CW approximation of $\varprojlim {}X_{n}$ . They can be constructed on a CW complex by iteratively killing off homotopy groups. If we have a map $f:S^{n}\to X$ representing a homotopy class $[f]\in \pi _{n}(X)$ , we can take the pushout along the boundary map $S^{n}\to e_{n+1}$ , killing off the homotopy class. For $X_{m}$ this process can be repeated for all $n>m$ , giving a space which has vanishing homotopy groups $\pi _{n}(X_{m})$ . Using the fact that $X_{n-1}$ can be constructed from $X_{n}$ by killing off all homotopy maps $S^{n}\to X_{n}$ , we get a map $X_{n}\to X_{n-1}$ .

Homotopy groups of spheres

One application of the Postnikov tower is the computation of homotopy groups of spheres. For $S^{n}$ we can use the Hurewicz theorem to show each $S_{i}^{n}$ is contractible for $i<n$ since the theorem implies the lower homotopy groups are trivial. Recall there is a spectral sequence for any Serre fibration, such as the fibration

$K(\pi _{n+1}(X),n+1)\simeq F_{n+1}\to S_{n+1}^{n}\to S_{n}^{n}\simeq K(\mathbb {Z} ,n)$

We can then form a homological spectral sequence with $E^{2}$ -terms

$E_{p,q}^{2}=H_{p}(K(\mathbb {Z} ,n),H_{q}(K(\pi _{n+1}(S^{n}),n+1)))$ .

And the first non-trivial map to $\pi _{n+1}(S^{n})$

$d_{0,n+1}^{n+1}:H_{n+2}(K(\mathbb {Z} ,n))\to H_{0}(K(\mathbb {Z} ,n),H_{n+1}(K(\pi _{n+1}(S^{n}),n+1)))$

equivalently written as

$d_{0,n+1}^{n+1}:H_{n+2}(K(\mathbb {Z} ,n))\to \pi _{n+1}(S^{n})$

If it's easy to compute $H_{n+1}(S_{n+1}^{n})$ and $H_{n+2}(S_{n+2}^{n})$ , then we can get information about what this map looks like. In particular, if it's an isomorphism, we get a computation of $\pi _{n+1}(S^{n})$ .

Whitehead tower

Given a CW complex $X$ there is a dual construction to the Postnikov tower called the Whitehead tower. Instead of killing off all higher homotopy groups, the Whitehead tower iteratively kills off lower homotopy groups. This is given by a tower of CW complexes

$\cdots \to X_{3}\to X_{2}\to X_{1}\to X$

where

(1) The lower homotopy groups are zero, so $\pi _{i}(X_{n})=0$ for $i\leq n$

(2) The induced map $\pi _{i}:\pi _{i}(X_{n})\to \pi _{i}(X)$ is an isomorphism for $i>n$

(3) The maps $X_{n}\to X_{n-1}$ are fibrations with fiber $K(\pi _{n}(X),n-1)$

Implications

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

Construction

The spaces $X_{n}$ in the Whitehead tower are constructed inductively. If we construct a $K(\pi _{n+1}(X),n+1)$ by killing off the higher homotopy groups in $X_{n}$ [3], we get an embedding $X_{n}\to K(\pi _{n+1}(X),n+1)$ . If we let

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

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

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

so we have a Serre fibration

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

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

$0\to \pi _{n+1}(X_{n+1})\to \pi _{n+1}(X_{n}){\xrightarrow {\partial }}\pi _{n}K(\pi _{n+1}(X),n)\to \pi _{n}(X_{n+1})\to 0$

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

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

so

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

giving the desired result.

Whitehead tower and string theory

In Spin geometry the $\operatorname {Spin} (n)$ group is constructed as the universal cover of the Special orthogonal group $\operatorname {SO} (n)$ , so $\mathbb {Z} /2\to \operatorname {Spin} (n)\to SO(n)$ is a fibration, giving the first term in the Whitehead tower. There are physically relevant interpretations for the higher parts in this tower, which can be read as

\cdots \to \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\to \operatorname {Spin} (n)\to \operatorname {SO} (n)

where $\operatorname {String} (n)$ is the $3$ -connected cover of $\operatorname {SO} (n)$ , and $\operatorname {Fivebrane} (n)$ is the $7$ -connected cover.[4][5]

References

Hatcher, Allen. Algebraic Topology (PDF). p. 410.
Hatcher, Allen. Algebraic Topology (PDF). p. 354.
Maxim, Laurentiu. "Lecture Notes on Homotopy Theory and Applications" (PDF). p. 66. Archived (PDF) from the original on 16 Feb 2020.
"Mathematical physics – Physical application of Postnikov tower, String(n) and Fivebrane(n)". Physics Stack Exchange. Retrieved 2020-02-16.
"at.algebraic topology – What do Whitehead towers have to do with physics?". MathOverflow. Retrieved 2020-02-16.

Postnikov, Mikhail M. (1951). "Determination of the homology groups of a space by means of the homotopy invariants". Doklady Akademii Nauk SSSR. 76: 359–362.
Lecture Notes on Homotopy Theory and Applications
Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. ISBN 978-0-521-79540-1.
Handwritten notes https://web.archive.org/web/20200213180540/https://www.math.purdue.edu/~zhang24/towers.pdf

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[1] Hatcher, Allen. Algebraic Topology (PDF). p. 410.

[2] Hatcher, Allen. Algebraic Topology (PDF). p. 354.

[3] Maxim, Laurentiu. "Lecture Notes on Homotopy Theory and Applications" (PDF). p. 66. Archived (PDF) from the original on 16 Feb 2020.

[4] "Mathematical physics – Physical application of Postnikov tower, String(n) and Fivebrane(n)". Physics Stack Exchange. Retrieved 2020-02-16.

[5] "at.algebraic topology – What do Whitehead towers have to do with physics?". MathOverflow. Retrieved 2020-02-16.