Postnikov system

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

: $\backslash cdots\; \backslash to\; X\_n\; \backslash xrightarrow\{p\_n\}\; X\_\{n-1\}\backslash xrightarrow\{p\_\{n-1\}\}\; \backslash cdots\; \backslash xrightarrow\{p\_3\}\; X\_2\; \backslash xrightarrow\{p\_2\}\; X\_1\; \backslash xrightarrow\{p\_1\}\; *$

with a sequence of maps $\backslash phi\_n\; :\; X\; \backslash to\; X\_n$ compatible with the inverse system such that

1. The map $\backslash phi\_n\; :\; X\; \backslash to\; X\_n$ induces an isomorphism $\backslash pi\_i(X)\; \backslash to\; \backslash pi\_i(X\_n)$ for every $i\backslash leq\; n$.
2. $\backslash pi\_i(X\_n)\; =\; 0$ for $i\; >\; n$.{{Cite book|last=Hatcher|first=Allen|author-link=Allen Hatcher|url=https://pi.math.cornell.edu/~hatcher/AT/AT.pdf|title=Algebraic Topology}}{{rp|410}}
3. Each map $p\_n\; :\; X\_n\; \backslash to\; X\_\{n-1\}$ is a fibration, and so the fiber $F\_n$ is an Eilenberg–MacLane space, $K(\backslash pi\_n(X),n)$.

The first two conditions imply that $X\_1$ is also a $K(\backslash pi\_1(X),1)$-space. More generally, if $X$ is $(n-1)$-connected, then $X\_n$ is a $K(\backslash pi\_n(X),n)$-space and all $X\_\{i\}$ for are contractible. Note the third condition is only included optionally by some authors.

Postnikov systems exist on connected CW complexes,{{rp|354}} and there is a weak homotopy-equivalence between $X$ and its inverse limit, so

: $X\backslash simeq\backslash varprojlim\{\}X\_n$,

showing that $X$ 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 $f\; :\; S^\{n\}\; \backslash to\; X$ representing a homotopy class $[f]\backslash in\backslash pi\_n(X)$, we can take the pushout along the boundary map $S^\{n\}\; \backslash to\; e\_\{n+1\}$, 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 $\backslash pi\_n(X\_m)$. Using the fact that $X\_\{n-1\}$can be constructed from $X\_n$ by killing off all homotopy maps $S^n\; \backslash to\; X\_\{n\}$, we obtain a map $X\_n\; \backslash to\; X\_\{n-1\}$.

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

The sequence of fibrations $p\_n:X\_n\; \backslash to\; X\_\{n-1\}${{Cite journal|last=Kahn|first=Donald W.|date=1963-03-01|title=Induced maps for Postnikov systems|url=https://www.ams.org/journals/tran/1963-107-03/S0002-9947-1963-0150777-X/S0002-9947-1963-0150777-X.pdf|journal=Transactions of the American Mathematical Society| volume=107|issue=3|pages=432–450|doi=10.1090/s0002-9947-1963-0150777-x|issn=0002-9947|doi-access=free}} have homotopically defined invariants, meaning the homotopy classes of maps $p\_n$, give a well defined homotopy type $[X]\; \backslash in\; \backslash operatorname\{Ob\}(hTop)$. The homotopy class of $p\_n$ comes from looking at the homotopy class of the classifying map for the fiber $K(\backslash pi\_n(X),\; n)$. The associated classifying map is

:$X\_\{n-1\}\; \backslash to\; B(K(\backslash pi\_n(X),n))\; \backslash simeq\; K(\backslash pi\_n(X),n+1)$,

hence the homotopy class $[p\_n]$ is classified by a homotopy class

:$[p\_n]\; \backslash in\; [X\_\{n-1\},K(\backslash pi\_n(X),\; n+1)]\; \backslash cong\; H^\{n+1\}(X\_\{n-1\},\; \backslash pi\_n(X))$

called the nth Postnikov invariant of $X$, since the homotopy classes of maps to Eilenberg-Maclane spaces gives cohomology with coefficients in the associated abelian group.

One of the special cases of the homotopy classification is the homotopy class of spaces $X$ such that there exists a fibration

:$K(A,n)\; \backslash to\; X\; \backslash to\; \backslash pi\_1(X)$

giving a homotopy type with two non-trivial homotopy groups, $\backslash pi\_1(X)\; =\; G$, and $\backslash pi\_n(X)\; =\; A$. Then, from the previous discussion, the fibration map $BG\; \backslash to\; K(A,n+1)$ gives a cohomology class in

:$H^\{n+1\}(BG,\; A)$,

which can also be interpreted as a group cohomology class. This space $X$ can be considered a higher local system.

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

:$\backslash begin\{matrix\}$

X_i \simeq * &\text{for } i < n \\

X_i \simeq K(G,n) & \text{for } i \geq n

\end{matrix}

The Postnikov tower for the sphere $S^2$ 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 $S^2$, degree theory of spheres, and the Hopf fibration, giving $\backslash pi\_k(S^2)\; \backslash simeq\; \backslash pi\_k(S^3)$ for $k\; \backslash geq\; 3$, hence

: $\backslash begin\{matrix\}$

\pi_1(S^2) =& 0 \\

\pi_2(S^2) =& \Z \\

\pi_3(S^2) =& \Z \\

\pi_4(S^2) =& \Z/2.

\end{matrix}

Then, $X\_2\; =\; S^2\_2\; =\; K(\backslash Z,2)$, and $X\_3$ comes from a pullback sequence

: $\backslash begin\{matrix\}$

X_3 & \to & * \\

\downarrow & & \downarrow \\

X_2 & \to & K(\Z,4) ,

\end{matrix}

which is an element in

: $[p\_3]\; \backslash in\; [K(\backslash Z,2),\; K(\backslash Z,4)]\; \backslash cong\; H^4(\backslash mathbb\{CP\}^\backslash infty)\; =\; \backslash Z$.

If this was trivial it would imply $X\_3\; \backslash simeq\; K(\backslash Z,2)\backslash times\; K(\backslash Z,3)$. But, this is not the case! In fact, this is responsible for why strict infinity groupoids don't model homotopy types.{{cite arXiv|last=Simpson|first=Carlos|author-link=Carlos Simpson|date=1998-10-09|title=Homotopy types of strict 3-groupoids|eprint=math/9810059}} Computing this invariant requires more work, but can be explicitly found.{{Cite journal|last1=Eilenberg|first1=Samuel|author1-link=Samuel Eilenberg|last2=MacLane|first2=Saunders|author2-link=Saunders MacLane|date=1954|title=On the Groups $H(\backslash Pi,\; n)$, III: Operations and Obstructions|jstor=1969849|journal=Annals of Mathematics|volume=60|issue=3|pages=513–557|doi=10.2307/1969849|issn=0003-486X}} This is the quadratic form $x\; \backslash mapsto\; x^2$ on $\backslash Z\; \backslash to\; \backslash Z$ coming from the Hopf fibration $S^3\; \backslash to\; S^2$. Note that each element in $H^4(\backslash mathbb\{CP\}^\backslash infty)$ gives a different homotopy 3-type.

One application of the Postnikov tower is the computation of homotopy groups of spheres.{{Cite web|last=Laurențiu-George|first=Maxim|title=Spectral sequences and homotopy groups of spheres|url=https://www.math.wisc.edu/~maxim/753f13w7.pdf|url-status=live|archive-url=https://web.archive.org/web/20170519125745/https://www.math.wisc.edu/~maxim/753f13w7.pdf|archive-date=19 May 2017}} For an $n$-dimensional sphere $S^n$ we can use the Hurewicz theorem to show each $S^n\_i$ is contractible for , since the theorem implies that the lower homotopy groups are trivial. Recall there is a spectral sequence for any Serre fibration, such as the fibration

: $K(\backslash pi\_\{n+1\}(X),\; n\; +\; 1)\; \backslash simeq\; F\_\{n+1\}\; \backslash to\; S^n\_\{n+1\}\; \backslash to\; S^n\_n\; \backslash simeq\; K(\backslash Z,\; n)$.

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

: $E^2\_\{p,q\}\; =\; H\_p\backslash left(K(\backslash Z,\; n),\; H\_q\backslash left(K\backslash left(\backslash pi\_\{n+1\}\backslash left(S^n\backslash right),\; n\; +\; 1\backslash right)\backslash right)\backslash right)$.

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

: $d^\{n+1\}\_\{0,n+1\}\; :\; H\_\{n+2\}(K(\backslash Z,\; n))\; \backslash to\; H\_0\backslash left(K(\backslash Z,\; n),\; H\_\{n+1\}\backslash left(K\backslash left(\backslash pi\_\{n+1\}\backslash left(S^n\backslash right),\; n\; +\; 1\backslash right)\backslash right)\backslash right)$,

equivalently written as

: $d^\{n+1\}\_\{0,n+1\}\; :\; H\_\{n+2\}(K(\backslash Z,\; n))\; \backslash to\; \backslash pi\_\{n+1\}\backslash left(S^n\backslash right)$.

If it's easy to compute $H\_\{n+1\}\backslash left(S^n\_\{n+1\}\backslash right)$ and $H\_\{n+2\}\backslash left(S^n\_\{n+2\}\backslash right)$, then we can get information about what this map looks like. In particular, if it's an isomorphism, we obtain a computation of $\backslash pi\_\{n+1\}\backslash left(S^n\backslash right)$. For the case $n\; =\; 3$, this can be computed explicitly using the path fibration for $K(\backslash Z,\; 3)$, the main property of the Postnikov tower for $\backslash mathfrak\{X\}\_4\; \backslash simeq\; S^3\; \backslash cup\; \backslash \{\backslash text\{cells\; of\; dimension\}\; \backslash geq\; 6\backslash \}$ (giving $H\_4(X\_4)\; =\; H\_5(X\_4)\; =\; 0$, and the universal coefficient theorem giving $\backslash pi\_4\backslash left(S^3\backslash right)\; =\; \backslash Z/2$. Moreover, because of the Freudenthal suspension theorem this actually gives the stable homotopy group $\backslash pi\_1^\backslash mathbb\{S\}$ since $\backslash pi\_\{n+k\}\backslash left(S^n\backslash right)$ is stable for $n\; \backslash geq\; k\; +\; 2$.

Note that similar techniques can be applied using the Whitehead tower (below) for computing $\backslash pi\_4\backslash left(S^3\backslash right)$ and $\backslash pi\_5\backslash left(S^3\backslash right)$, giving the first two non-trivial stable homotopy groups of spheres.

In addition to the classical Postnikov tower, there is a notion of Postnikov towers in stable homotopy theory constructed on spectra{{Cite book|url=http://link.springer.com/10.1007/978-3-540-77751-9|title=On Thom Spectra, Orientability, and Cobordism|date=1998|publisher=Springer|isbn=978-3-540-62043-3|series=Springer Monographs in Mathematics|location=Berlin, Heidelberg|language=en|doi=10.1007/978-3-540-77751-9}}^{pg 85-86}.

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

: $\backslash cdots\; \backslash to\; E\_\{(2)\}\; \backslash xrightarrow\{p\_2\}\; E\_\{(1)\}\; \backslash xrightarrow\{p\_1\}\; E\_\{(0)\}$,

with maps

: $\backslash tau\_n\; :\; E\; \backslash to\; E\_\{(n)\}$

commuting with the $p\_n$ maps. Then, this tower is a Postnikov tower if the following two conditions are satisfied:

1. $\backslash pi\_i^\{\backslash mathbb\{S\}\}\backslash left(E\_\{(n)\}\backslash right)\; =\; 0$ for $i\; >\; n$,
2. $\backslash left(\backslash tau\_n\backslash right)\_*\; :\; \backslash pi\_i^\{\backslash mathbb\{S\}\}(E)\; \backslash to\; \backslash pi\_i^\{\backslash mathbb\{S\}\}\backslash left(E\_\{(n)\}\backslash right)$ is an isomorphism for $i\; \backslash leq\; n$,

where $\backslash pi\_i^\{\backslash mathbb\{S\}\}$ 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.

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,

: $\backslash cdots\; \backslash to\; X\_3\; \backslash to\; X\_2\; \backslash to\; X\_1\; \backslash to\; X$,

where

1. The lower homotopy groups are zero, so $\backslash pi\_i(X\_n)\; =\; 0$ for $i\; \backslash leq\; n$.
2. The induced map $\backslash pi\_i\; :\; \backslash pi\_i(X\_n)\; \backslash to\; \backslash pi\_i(X)$ is an isomorphism for $i\; >\; n$.
3. The maps $X\_n\; \backslash to\; X\_\{n-1\}$ are fibrations with fiber $K(\backslash pi\_n(X),\; n-1)$.

Notice $X\_1\; \backslash to\; X$ is the universal cover of $X$ since it is a covering space with a simply connected cover. Furthermore, each $X\_n\; \backslash to\; X$ is the universal $n$-connected cover of $X$.

The spaces $X\_n$ in the Whitehead tower are constructed inductively. If we construct a $K\backslash left(\backslash pi\_\{n+1\}(X),\; n\; +\; 1\backslash right)$ by killing off the higher homotopy groups in $X\_n$,{{Cite web|url=https://www.math.wisc.edu/~maxim/754notes.pdf|title=Lecture Notes on Homotopy Theory and Applications|first=Laurențiu|last=Maxim|page=66|url-status=live|archive-url=https://web.archive.org/web/20200216062602/https://www.math.wisc.edu/~maxim/754notes.pdf|archive-date=16 February 2020}} we get an embedding $X\_n\; \backslash to\; K(\backslash pi\_\{n+1\}(X),\; n\; +\; 1)$. If we let

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

for some fixed basepoint $p$, then the induced map $X\_\{n+1\}\; \backslash to\; X\_n$ is a fiber bundle with fiber homeomorphic to

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

and so we have a Serre fibration

: $K\backslash left(\backslash pi\_\{n+1\}(X),\; n\backslash right)\; \backslash to\; X\_n\; \backslash to\; X\_\{n-1\}$.

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

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

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

: $X\_\{n-1\}\; \backslash cup\; \backslash \{\backslash text\{cells\; of\; dimension\}\; \backslash geq\; n\; +\; 2\backslash \}$; thus,

: $\backslash pi\_\{n+1\}\backslash left(X\_n\backslash right)\; \backslash cong\; \backslash pi\_\{n+1\}\backslash left(K\backslash left(\backslash pi\_\{n+1\}(X),\; n\; +\; 1\backslash right)\backslash right)\; \backslash cong\; \backslash pi\_n\backslash left(K\backslash left(\backslash pi\_\{n+1\}(X),\; n\backslash right)\backslash right)$,

giving the desired result.

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

: $\backslash text\{Hofiber\}(\backslash phi\_n:\; X\; \backslash to\; X\_n)$

from the Postnikov tower, we get a space $X^n$ which has

: $\backslash pi\_k(X^n)\; =\; \backslash begin\{cases\}$

\pi_k(X) & k > n \\

0 & k \leq n

\end{cases}

The dual notion of the Whitehead tower can be defined in a similar manner using homotopy fibers in the category of spectra. If we let

: $E\backslash langle\; n\; \backslash rangle\; =\; \backslash operatorname\{Hofiber\}\backslash left(\backslash tau\_n:\; E\; \backslash to\; E\_\{(n)\}\backslash right)$

then this can be organized in a tower giving connected covers of a spectrum. This is a widely used construction{{Cite journal|last=Hill|first=Michael A.|date=2009|title=The string bordism of BE₈ and BE₈ × BE₈ through dimension 14|url=https://projecteuclid.org/euclid.ijm/1264170845|journal=Illinois Journal of Mathematics|language=EN|volume=53|issue=1|pages=183–196|doi=10.1215/ijm/1264170845|issn=0019-2082|doi-access=free}}{{Cite journal|date=2014-12-01|title=Secondary invariants for string bordism and topological modular forms|journal=Bulletin des Sciences Mathématiques|language=en|volume=138|issue=8|pages=912–970|doi=10.1016/j.bulsci.2014.05.002|issn=0007-4497|doi-access=free|last1=Bunke|first1=Ulrich|last2=Naumann|first2=Niko}}{{cite book|last=Szymik|first=Markus|year=2019|chapter=String bordism and chromatic characteristics|editor1=Daniel G. Davis|editor2=Hans-Werner Henn|editor3=J. F. Jardine|editor4=Mark W. Johnson|editor5=Charles Rezk|title=Homotopy Theory: Tools and Applications|series=Contemporary Mathematics|volume=729|pages=239–254|doi=10.1090/conm/729/14698|arxiv=1312.4658|isbn=9781470442446|s2cid=56461325}} in bordism theory because the coverings of the unoriented cobordism spectrum $M\backslash text\{O\}$ gives other bordism theories

: $\backslash begin\{align\}$

M\text{String} &= M\text{O}\langle 8 \rangle \\

M\text{Spin} &= M\text{O}\langle 4 \rangle \\

M\text{SO} &= M\text{O}\langle 2 \rangle

\end{align}

such as string bordism.

In Spin geometry the $\backslash operatorname\{Spin\}(n)$ group is constructed as the universal cover of the Special orthogonal group $\backslash operatorname\{SO\}(n)$, so $\backslash Z/2\; \backslash to\; \backslash operatorname\{Spin\}(n)\; \backslash to\; SO(n)$ 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

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

• Adams spectral sequence
• Eilenberg–MacLane space
• CW complex
• Obstruction theory
• Stable homotopy theory
• Homotopy groups of spheres
• Higher group
• Hopf–Whitney theorem, application to calculate homotopy classes

{{reflist}}

• {{cite journal | last=Postnikov | first=Mikhail M. | author-link=Mikhail Postnikov | title=Determination of the homology groups of a space by means of the homotopy invariants | journal=Doklady Akademii Nauk SSSR | volume=76 | year=1951 | pages=359–362}}
• {{cite web|last=Maxim|first=Laurențiu|url=https://web.archive.org/web/20200216062602/https://www.math.wisc.edu/~maxim/754notes.pdf| title=Lecture Notes on Homotopy Theory and Applications|website=www.math.wisc.edu}}
• [https://www.pnas.org/content/32/11/277 Determination of the Second Homology and Cohomology Groups of a Space by Means of Homotopy Invariants] - gives accessible examples of Postnikov invariants
• {{Cite book | last=Hatcher | first=Allen | author-link=Allen Hatcher|title=Algebraic topology | url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=Cambridge University Press | isbn=978-0-521-79540-1 | year=2002}}
• {{cite web|author=Zhang|title=Postnikov towers, Whitehead towers and their applications (handwritten notes)|url=https://www.math.purdue.edu/~zhang24/towers.pdf |archive-url=https://web.archive.org/web/20200213180540/https://www.math.purdue.edu/~zhang24/towers.pdf |website=www.math.purdue.edu|url-status=dead |archive-date=2020-02-13 }}

Category:Homotopy theory