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Difference between revisions of "Postnikov system"

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''natural system, homotopic resolution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p0740802.png" />-decomposition of general type''
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{{TEX|done}}
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''natural system, homotopic resolution, $P$-decomposition of general type''
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$\newcommand{\too}{\longrightarrow}$
  
 
A sequence of fibrations
 
A sequence of fibrations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p0740803.png" /></td> </tr></table>
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$$\cdots \xrightarrow{p_{n+1}} X_n \xrightarrow{\ p_n\ } X_{n-1} \xrightarrow{p_{n-1}} \cdots \xrightarrow{\ p_1\ } X_0 = \text{pt},$$
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whose fibres are the Eilenberg–MacLane spaces $K(\pi_n, n)$ (cf.
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[[Eilenberg–MacLane space|Eilenberg–MacLane space]]), where $\pi_n$ is some group (Abelian for $n > 1$). This system was introduced by M.M. Postnikov
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[[#References|[1]]]. The space $X_n$ is called the $n$-th term (or the $n$-th layer) of the Postnikov system $\{p_n : X_n \to X_{n-1} \}$. The Postnikov system $\{p_n : X_n \to X_{n-1}\}$ is said to converge to a space $X$ if its inverse limit $\varprojlim \{p_n:X_n \to X_{n-1}\}$ is weakly homotopy equivalent to $X$. In this case $X$ is called the limit of the Postnikov system $\{p_n: X_n \to X_{n-1}\}$.
  
whose fibres are the Eilenberg–MacLane spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p0740804.png" /> (cf. [[Eilenberg–MacLane space|Eilenberg–MacLane space]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p0740805.png" /> is some group (Abelian for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p0740806.png" />). This system was introduced by M.M. Postnikov [[#References|[1]]]. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p0740807.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p0740808.png" />-th term (or the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p0740809.png" />-th layer) of the Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408010.png" />. The Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408011.png" /> is said to converge to a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408012.png" /> if its inverse limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408013.png" /> is weakly homotopy equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408014.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408015.png" /> is called the limit of the Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408016.png" />.
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A morphism of a Postnikov system $\{p_n : X_n \to X_{n-1}\}$ into a Postnikov system $\{q_n: Y_n \to Y_{n-1} \}$ is a sequence of continuous mappings $f_n : X_n \to Y_n$ such that the diagram below is homotopy commutative. A morphism $\{f_n\}$ induces a mapping $\varprojlim f_n : \varprojlim X_n \to \varprojlim Y_n$, which is called its limit.
  
A morphism of a Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408017.png" /> into a Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408018.png" /> is a sequence of continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408019.png" /> such that the diagram below is homotopy commutative. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408020.png" /> induces a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408021.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408022.png" />, which is called its limit.
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\begin{array}{ccccccccc}
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\cdots & \too & X_n & \xrightarrow{p_n} & X_{n-1} & \too & \cdots & \too & X_1\\
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& & \big\downarrow f_n & & \big\downarrow f_{n-1} & & & & \big\downarrow f_1\\
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\cdots & \too & Y_n & \xrightarrow{p_n} & Y_{n-1} & \too & \cdots & \too & Y_1
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\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408023.png" /></td> </tr></table>
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The definition of a Postnikov system implies that for any $n \ge 1$ the mapping $p_n$ is an $(n-1)$-equivalence (see
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[[Homotopy type|Homotopy type]]). In particular, $\pi_i(X_{n-1}) \cong \pi_1(X_n)$ for $i < n$, $\pi_n(X_n) \cong \pi_n$ and $\pi_(X_n) = 0$ for $i > n$. The spaces $X$ and $X_n$ are of the same $(n+1)$-type. In particular, if the Postnikov system is finite, i.e. if for some number $N$ for all $n > N$ the group $\pi_n$ is trivial, then $X$ and $X_n$ are homotopy equivalent. In the general case, for $i \le n$ there are isomorphisms $H_i(X_n) \cong H_i(X)$ and $\pi_i(X_n) \cong\pi_i(X)$, i.e. the homology groups and the homotopy groups stabilize when $n$ tends to infinity. For any CW-complex $K$ of dimension $\le n$ the sets $[K, X]$ and $[K, X_n]$ coincide. The characteristic class $k_n = c(p_n) \in H^{n+1}(X_{n-1}; \{\pi_n\})$ of the fibration $p_n: X_n \to X_{n-1}$, i.e. the image under the
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[[Transgression|transgression]]
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$$\tau : H^n(K(\pi, n); \pi) \to H^{n+1}(B; \{\pi\})$$
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of the
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[[Fundamental class|fundamental class]] $\iota_n \in H^n(K(\pi, n); \pi)$, is called the $n$-th $K$-invariant (or the $n$-th Postnikov factor) of the Postnikov system or of its limit $X$. For any $n \ge 1$ the $n$-th term of the Postnikov system, and hence the $(n+1)$-type of $X$, are completely determined by the groups $\pi_1, \ldots, \pi_n$ and the $K$-invariants $k_1, \ldots, k_{n-1}$. Often the double sequence below is called a Postnikov system:
  
The definition of a Postnikov system implies that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408024.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408025.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408026.png" />-equivalence (see [[Homotopy type|Homotopy type]]). In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408027.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408030.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408031.png" />. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408033.png" /> are of the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408034.png" />-type. In particular, if the Postnikov system is finite, i.e. if for some number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408035.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408036.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408037.png" /> is trivial, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408039.png" /> are homotopy equivalent. In the general case, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408040.png" /> there are isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408042.png" />, i.e. the homology groups and the homotopy groups stabilize when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408043.png" /> tends to infinity. For any CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408044.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408045.png" /> the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408047.png" /> coincide. The characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408048.png" /> of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408049.png" />, i.e. the image under the [[Transgression|transgression]]
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$$\{\pi_1, k_1, \ldots, \pi_n, k_n, \ldots \}.$$
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A space $X$ is the limit of a Postnikov system $\{p_n : X_n \to X_{n-1}\}$ if and only if there exist $(n-1)$-equivalences $\rho_n : X \to X_n$ such that $\rho_{n-1} \sim p_n \circ \rho_n$ for any $n \ge 1$. Limits of morphisms of a Postnikov system are characterized analogously.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408050.png" /></td> </tr></table>
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There exists a version of the notion of a Postnikov system which sometimes turns out to be more useful. In this version the spaces $X_n$ are assumed to be CW-complexes such that $X_{n-1} \subset X_n^n$ and $X_{n-1}^{n-1} = X_n^{n-1}$, and the mappings $p_n: X_n \to X_{n-1}$ are taken to be cellular mappings (which are not fibrations any more) such that, first, $p_n|_{X_n^{n-1}} = \text{id}$ and, secondly, the homotopy fibre of the mapping $p_n$ (i.e. the fibre of this mapping turned into a fibration) is the space $K(\pi_n, n)$. Such Postnikov systems are called cellular. The limit of a cellular Postnikov system is a CW-complex $X$ for which $X^n = X_n^n$ for any $n \ge 1$. An arbitrary Postnikov system is homotopy equivalent to a cellular Postnikov system.
  
of the [[Fundamental class|fundamental class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408051.png" />, is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408052.png" />-th <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408054.png" />-invariant (or the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408055.png" />-th Postnikov factor) of the Postnikov system or of its limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408056.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408057.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408058.png" />-th term of the Postnikov system, and hence the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408059.png" />-type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408060.png" />, are completely determined by the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408061.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408062.png" />-invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408063.png" />. Often the double sequence below is called a Postnikov system:
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The fundamental theorem in the theory of Postnikov systems states (see
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[[#References|[1]]],
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[[#References|[6]]]) that each space $X$ is the limit of some unique (up to isomorphism) Postnikov system $\{p_n: X_n \to X_{n-1}\}$. This Postnikov system is called the Postnikov system of the space $X$. A version of the fundamental theorem for mappings holds: Any mapping $f : X \to Y$ is the limit of some morphism $\{f_n : X_n \to Y_n\}$ of the Postnikov system $\{p_n : X_n \to X_{n-1}\}$ of $X$ into the Postnikov system $\{q_n : Y_n \to Y_{n-1}\}$ of $Y$. This morphism is called the Postnikov system of the mapping $f$ (it is also called the homotopic resolution, the $P$-system of general type or the Moore–Postnikov system of the mapping). For a constant mapping $c : X \to \text{pt}$ of a path-connected space $X$ its Postnikov system coincides with the Postnikov system of the space $X$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408064.png" /></td> </tr></table>
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In applications the so-called standard Postnikov systems (often called just Postnikov systems) are widely spread. These systems are Postnikov systems which consist of the principal fibrations $p_n: X_n \to X_{n-1}$ induced from the standard Serre fibrations $K(\pi_n, n) \to EK(\pi_n, n+1) \to K(\pi_n, n+1)$ by the Postnikov factors $k_n \in H^{n+1}(X_{n-1}; \pi_n)$ interpreted as mappings $k_n: X_{n-1} \to K(\pi_n, n+1)$ by virtue of the representation of the cohomology group as $H^{n+1}(X_{n-1}; \pi_n) \cong [X_{n-1}, K(\pi_n, n+1)]$. All spaces which are homotopy simple in all dimensions (Abelian spaces in the terminology of
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[[#References|[2]]]) and only these spaces have standard Postnikov systems (see
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[[#References|[3]]],
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[[#References|[4]]]).
  
A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408065.png" /> is the limit of a Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408066.png" /> if and only if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408067.png" />-equivalences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408069.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408070.png" />. Limits of morphisms of a Postnikov system are characterized analogously.
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Standard Postnikov systems are applied to solve extension and lifting problems to which numerous problems in algebraic topology are reduced. The combined formulation of these problems is the following. Let a (homotopy) commutative square of spaces and mappings be given in which the mapping $i$ is a closed cofibration with cofibre $X/A$ and $p$ is a fibration with fibre $F$. The question is whether there exists a mapping $X\to Y$ such that both triangles obtained are (homotopy) commutative.
  
There exists a version of the notion of a Postnikov system which sometimes turns out to be more useful. In this version the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408071.png" /> are assumed to be CW-complexes such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408073.png" />, and the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408074.png" /> are taken to be cellular mappings (which are not fibrations any more) such that, first, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408075.png" /> and, secondly, the homotopy fibre of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408076.png" /> (i.e. the fibre of this mapping turned into a fibration) is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408077.png" />. Such Postnikov systems are called cellular. The limit of a cellular Postnikov system is a CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408078.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408079.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408080.png" />. An arbitrary Postnikov system is homotopy equivalent to a cellular Postnikov system.
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\begin{array}{ccc}
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A & \too & Y\\
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i\big \downarrow & & \big \downarrow p\\
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X & \too & B
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\end{array}
  
The fundamental theorem in the theory of Postnikov systems states (see [[#References|[1]]], [[#References|[6]]]) that each space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408081.png" /> is the limit of some unique (up to isomorphism) Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408082.png" />. This Postnikov system is called the Postnikov system of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408083.png" />. A version of the fundamental theorem for mappings holds: Any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408084.png" /> is the limit of some morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408085.png" /> of the Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408086.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408087.png" /> into the Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408088.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408089.png" />. This morphism is called the Postnikov system of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408090.png" /> (it is also called the homotopic resolution, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408092.png" />-system of general type or the Moore–Postnikov system of the mapping). For a constant mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408093.png" /> of a path-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408094.png" /> its Postnikov system coincides with the Postnikov system of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408095.png" />.
+
Further, if such a mapping does exist, then one is expected to determine the set $[X, Y]_B^A$ of homotopy classes of mappings $X \to Y$  "below A" (i.e. $\text{re } A$) and  "above B" . Suppose that for the fibration $p : Y \to B$ there exists a standard Postnikov system $\{p_n: Y_n \to Y_{n-1},\, Y_0 = B\}$ (for this purpose, for example, it is sufficient for the spaces $Y$ and $B$ to be simply connected). The problem of relative lifting is solved step by step.
  
In applications the so-called standard Postnikov systems (often called just Postnikov systems) are widely spread. These systems are Postnikov systems which consist of the principal fibrations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408096.png" /> induced from the standard Serre fibrations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408097.png" /> by the Postnikov factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408098.png" /> interpreted as mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p07408099.png" /> by virtue of the representation of the cohomology group as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080100.png" />. All spaces which are homotopy simple in all dimensions (Abelian spaces in the terminology of [[#References|[2]]]) and only these spaces have standard Postnikov systems (see [[#References|[3]]], [[#References|[4]]]).
+
Consider the "elementary" problem of the relative lifting of a mapping $f_{n-1}: X \to Y_{n-1}$ from the $(n-1)$-st term of the Postnikov system to its $n$-th term:
  
Standard Postnikov systems are applied to solve extension and lifting problems to which numerous problems in algebraic topology are reduced. The combined formulation of these problems is the following. Let a (homotopy) commutative square of spaces and mappings be given in which the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080101.png" /> is a closed cofibration with cofibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080103.png" /> is a fibration with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080104.png" />. The question is whether there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080105.png" /> such that both triangles obtained are (homotopy) commutative.
+
\begin{array}{ccccc}
 +
A & \xrightarrow{g_{n-1}} & Y_n & \too & EK(\pi_n(F), n+1)\\
 +
i \big \downarrow & & \big\downarrow p_n & & \big\downarrow\\
 +
X & \xrightarrow[f_{n-1}]{} & Y_{n-1} & \xrightarrow[k_n]{} & K(\pi_n(F), n+1)
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080106.png" /></td> </tr></table>
+
The mappings $f_{n-1}$ and $g_{n-1}$ define a mapping $X/A \to K(\pi_n(F), n+1)$, i.e. a cohomology class $c^{n+1} \in H^{n+1}(X, A, \pi_n(F))$, called an obstruction. The mapping $f_{n-1}$ can be lifted to $Y_n$ if and only if $c^{n+1} = 0$. Two liftings $f_n'$ and $f_n''$ determine an element $d^n \in H^n(X, A; \pi_n(F))$, called a difference, which is equal to zero if and only if the liftings $f_n'$ and $f_n''$ are homotopic.
  
Further, if such a mapping does exist, then one is expected to determine the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080107.png" /> of homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080108.png" />  "below A(i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080109.png" />) and  "above B" . Suppose that for the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080110.png" /> there exists a standard Postnikov system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080111.png" /> (for this purpose, for example, it is sufficient for the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080113.png" /> to be simply connected). The problem of relative lifting is solved step by step.
+
Thus, the problem of relative lifting is solved if the sequentially occurring obstructions $c^{n+1}$ vanish (for example, if $H^{n+1}(X, A; \pi_n(F)) = 0$). A lifting is unique if the sequentially occurring differences $d^n$ vanish (for example, if $H^n(X, A; \pi_n(F)) = 0$). In the case when the cofibration is an imbedding of CW-complexes, the obstruction $c^{n+1}$ and the difference $d^n$ coincide with usual  "cell-wise"  
 +
[[Obstruction|obstruction]] and difference (cf.
 +
[[Difference cochain and chain|Difference cochain and chain]]).
  
Consider the  "elementary"  problem of the relative lifting of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080114.png" /> from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080115.png" />-st term of the Postnikov system to its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080116.png" />-th term:
+
For simply-connected spaces $X$ with finitely-generated homology groups the Postnikov system is effectively computable
 
+
[[#References|[5]]] and, hence, the homotopy type of $X$ is effectively computable as well. However, in practice, for the majority of spaces one succeeds to compute only initial segments of Postnikov systems, which is due to the sharply increasing complexity of the computations. For computations one uses the method of cohomology operations (cf.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080117.png" /></td> </tr></table>
+
[[Cohomology operation|Cohomology operation]]).
 
 
The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080119.png" /> define a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080120.png" />, i.e. a cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080121.png" />, called an obstruction. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080122.png" /> can be lifted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080123.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080124.png" />. Two liftings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080126.png" /> determine an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080127.png" />, called a difference, which is equal to zero if and only if the liftings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080128.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080129.png" /> are homotopic.
 
 
 
Thus, the problem of relative lifting is solved if the sequentially occurring obstructions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080130.png" /> vanish (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080131.png" />). A lifting is unique if the sequentially occurring differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080132.png" /> vanish (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080133.png" />). In the case when the cofibration is an imbedding of CW-complexes, the obstruction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080134.png" /> and the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080135.png" /> coincide with usual  "cell-wise"  [[Obstruction|obstruction]] and difference (cf. [[Difference cochain and chain|Difference cochain and chain]]).
 
 
 
For simply-connected spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080136.png" /> with finitely-generated homology groups the Postnikov system is effectively computable [[#References|[5]]] and, hence, the homotopy type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080137.png" /> is effectively computable as well. However, in practice, for the majority of spaces one succeeds to compute only initial segments of Postnikov systems, which is due to the sharply increasing complexity of the computations. For computations one uses the method of cohomology operations (cf. [[Cohomology operation|Cohomology operation]]).
 
  
 
The dual of the Postnikov system is the Cartan–Serre system
 
The dual of the Postnikov system is the Cartan–Serre system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080138.png" /></td> </tr></table>
+
$$\cdots \to X_n^{CS} \to X_{n-1}^{CS} \to \cdots \to X_{k-1}^{CS} = X$$
  
of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080139.png" />, consisting of fibrations whose fibres are the Eilenberg–MacLane spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080140.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080141.png" /> is called the (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080142.png" />)-st [[Killing space|killing space]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080143.png" />. The terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080144.png" /> of the Cartan–Serre system are homotopy fibres of (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080145.png" />)-equivalences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080146.png" /> for the Postnikov system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080147.png" />, and the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080148.png" /> of a Postnikov system are loop spaces over the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080149.png" />.
+
of a space $X$, consisting of fibrations whose fibres are the Eilenberg–MacLane spaces $K(\pi_n(X), n-1)$. The space $X_n^{CS}$ is called the $(n+1)$-st
 +
[[Killing space|killing space]] for $X$. The terms $X_n^{CS}$ of the Cartan–Serre system are homotopy fibres of ($n-1$)-equivalences $\rho_n : X \to X_n$ for the Postnikov system of $X$, and the terms $X_n$ of a Postnikov system are loop spaces over the fibres of $X_n^{CS} \to X$.
  
 
A split Postnikov system is a sequence of principal fibrations
 
A split Postnikov system is a sequence of principal fibrations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080150.png" /></td> </tr></table>
+
$$\cdots \to X_n \to X_{n-1} \to \cdots \to X_0 = \text{pt}$$
 
+
whose fibres are the Eilenberg–MacLane spaces $K(\pi_n, s_n)$, $s_n \le s_{n+1}$. Split Postnikov systems are the principal tool in the studies of so-called nilpotent spaces and, in particular, of their localizations (see
whose fibres are the Eilenberg–MacLane spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074080/p074080152.png" />. Split Postnikov systems are the principal tool in the studies of so-called nilpotent spaces and, in particular, of their localizations (see [[Localization in categories|Localization in categories]], [[#References|[2]]], [[#References|[6]]], [[#References|[7]]]). There also exist other versions of Postnikov systems (see [[#References|[6]]]).
+
[[Localization in categories|Localization in categories]],
 +
[[#References|[2]]],
 +
[[#References|[6]]],
 +
[[#References|[7]]]). There also exist other versions of Postnikov systems (see
 +
[[#References|[6]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.M. Postnikov,  "Studies on the homotopy theory of continuous mappings" , '''1–2''' , Moscow  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.M. Postnikov,  "Localization of topological spaces"  ''Russian Math. Surveys'' , '''32''' :  6  (1977)  pp. 121–184  ''Uspekhi Mat. Nauk'' , '''32''' :  6  (1977)  pp. 117–181</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)  pp. Chapt. 13</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.H. Brown,  "Finite computability of Postnikov complexes"  ''Ann. of Math. (2)'' , '''65'''  (1957)  pp. 1–20</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H.J. Baues,  "Obstruction theory of homotopy classification of maps" , Springer  (1977)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P. Hilton,  G. Mislin,  J. Roitberg,  "Localization of nilpotent groups and spaces" , North-Holland  (1975)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD>
 +
<TD valign="top">  M.M. Postnikov,  "Studies on the homotopy theory of continuous mappings" , '''1–2''' , Moscow  (1955)  (In Russian)</TD>
 +
</TR><TR><TD valign="top">[2]</TD>
 +
<TD valign="top">  M.M. Postnikov,  "Localization of topological spaces"  ''Russian Math. Surveys'' , '''32''' :  6  (1977)  pp. 121–184  ''Uspekhi Mat. Nauk'' , '''32''' :  6  (1977)  pp. 117–181</TD>
 +
</TR><TR><TD valign="top">[3]</TD>
 +
<TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)  pp. Chapt. 13</TD>
 +
</TR><TR><TD valign="top">[4]</TD>
 +
<TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD>
 +
</TR><TR><TD valign="top">[5]</TD>
 +
<TD valign="top">  E.H. Brown,  "Finite computability of Postnikov complexes"  ''Ann. of Math. (2)'' , '''65'''  (1957)  pp. 1–20</TD>
 +
</TR><TR><TD valign="top">[6]</TD>
 +
<TD valign="top">  H.J. Baues,  "Obstruction theory of homotopy classification of maps" , Springer  (1977)</TD>
 +
</TR><TR><TD valign="top">[7]</TD>
 +
<TD valign="top">  P. Hilton,  G. Mislin,  J. Roitberg,  "Localization of nilpotent groups and spaces" , North-Holland  (1975)</TD>
 +
</TR></table>
  
  
Line 64: Line 107:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)  pp. Chapt. IX</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Gray,  "Homotopy theory. An introduction to algebraic topology" , Acad. Press  (1975)  pp. Chapt. 17</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD>
 +
<TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)  pp. Chapt. IX</TD>
 +
</TR><TR><TD valign="top">[a2]</TD>
 +
<TD valign="top">  B. Gray,  "Homotopy theory. An introduction to algebraic topology" , Acad. Press  (1975)  pp. Chapt. 17</TD>
 +
</TR></table>

Latest revision as of 07:35, 8 August 2018

natural system, homotopic resolution, $P$-decomposition of general type $\newcommand{\too}{\longrightarrow}$

A sequence of fibrations

$$\cdots \xrightarrow{p_{n+1}} X_n \xrightarrow{\ p_n\ } X_{n-1} \xrightarrow{p_{n-1}} \cdots \xrightarrow{\ p_1\ } X_0 = \text{pt},$$ whose fibres are the Eilenberg–MacLane spaces $K(\pi_n, n)$ (cf. Eilenberg–MacLane space), where $\pi_n$ is some group (Abelian for $n > 1$). This system was introduced by M.M. Postnikov [1]. The space $X_n$ is called the $n$-th term (or the $n$-th layer) of the Postnikov system $\{p_n : X_n \to X_{n-1} \}$. The Postnikov system $\{p_n : X_n \to X_{n-1}\}$ is said to converge to a space $X$ if its inverse limit $\varprojlim \{p_n:X_n \to X_{n-1}\}$ is weakly homotopy equivalent to $X$. In this case $X$ is called the limit of the Postnikov system $\{p_n: X_n \to X_{n-1}\}$.

A morphism of a Postnikov system $\{p_n : X_n \to X_{n-1}\}$ into a Postnikov system $\{q_n: Y_n \to Y_{n-1} \}$ is a sequence of continuous mappings $f_n : X_n \to Y_n$ such that the diagram below is homotopy commutative. A morphism $\{f_n\}$ induces a mapping $\varprojlim f_n : \varprojlim X_n \to \varprojlim Y_n$, which is called its limit.

\begin{array}{ccccccccc} \cdots & \too & X_n & \xrightarrow{p_n} & X_{n-1} & \too & \cdots & \too & X_1\\ & & \big\downarrow f_n & & \big\downarrow f_{n-1} & & & & \big\downarrow f_1\\ \cdots & \too & Y_n & \xrightarrow{p_n} & Y_{n-1} & \too & \cdots & \too & Y_1 \end{array}

The definition of a Postnikov system implies that for any $n \ge 1$ the mapping $p_n$ is an $(n-1)$-equivalence (see Homotopy type). In particular, $\pi_i(X_{n-1}) \cong \pi_1(X_n)$ for $i < n$, $\pi_n(X_n) \cong \pi_n$ and $\pi_(X_n) = 0$ for $i > n$. The spaces $X$ and $X_n$ are of the same $(n+1)$-type. In particular, if the Postnikov system is finite, i.e. if for some number $N$ for all $n > N$ the group $\pi_n$ is trivial, then $X$ and $X_n$ are homotopy equivalent. In the general case, for $i \le n$ there are isomorphisms $H_i(X_n) \cong H_i(X)$ and $\pi_i(X_n) \cong\pi_i(X)$, i.e. the homology groups and the homotopy groups stabilize when $n$ tends to infinity. For any CW-complex $K$ of dimension $\le n$ the sets $[K, X]$ and $[K, X_n]$ coincide. The characteristic class $k_n = c(p_n) \in H^{n+1}(X_{n-1}; \{\pi_n\})$ of the fibration $p_n: X_n \to X_{n-1}$, i.e. the image under the transgression $$\tau : H^n(K(\pi, n); \pi) \to H^{n+1}(B; \{\pi\})$$ of the fundamental class $\iota_n \in H^n(K(\pi, n); \pi)$, is called the $n$-th $K$-invariant (or the $n$-th Postnikov factor) of the Postnikov system or of its limit $X$. For any $n \ge 1$ the $n$-th term of the Postnikov system, and hence the $(n+1)$-type of $X$, are completely determined by the groups $\pi_1, \ldots, \pi_n$ and the $K$-invariants $k_1, \ldots, k_{n-1}$. Often the double sequence below is called a Postnikov system:

$$\{\pi_1, k_1, \ldots, \pi_n, k_n, \ldots \}.$$ A space $X$ is the limit of a Postnikov system $\{p_n : X_n \to X_{n-1}\}$ if and only if there exist $(n-1)$-equivalences $\rho_n : X \to X_n$ such that $\rho_{n-1} \sim p_n \circ \rho_n$ for any $n \ge 1$. Limits of morphisms of a Postnikov system are characterized analogously.

There exists a version of the notion of a Postnikov system which sometimes turns out to be more useful. In this version the spaces $X_n$ are assumed to be CW-complexes such that $X_{n-1} \subset X_n^n$ and $X_{n-1}^{n-1} = X_n^{n-1}$, and the mappings $p_n: X_n \to X_{n-1}$ are taken to be cellular mappings (which are not fibrations any more) such that, first, $p_n|_{X_n^{n-1}} = \text{id}$ and, secondly, the homotopy fibre of the mapping $p_n$ (i.e. the fibre of this mapping turned into a fibration) is the space $K(\pi_n, n)$. Such Postnikov systems are called cellular. The limit of a cellular Postnikov system is a CW-complex $X$ for which $X^n = X_n^n$ for any $n \ge 1$. An arbitrary Postnikov system is homotopy equivalent to a cellular Postnikov system.

The fundamental theorem in the theory of Postnikov systems states (see [1], [6]) that each space $X$ is the limit of some unique (up to isomorphism) Postnikov system $\{p_n: X_n \to X_{n-1}\}$. This Postnikov system is called the Postnikov system of the space $X$. A version of the fundamental theorem for mappings holds: Any mapping $f : X \to Y$ is the limit of some morphism $\{f_n : X_n \to Y_n\}$ of the Postnikov system $\{p_n : X_n \to X_{n-1}\}$ of $X$ into the Postnikov system $\{q_n : Y_n \to Y_{n-1}\}$ of $Y$. This morphism is called the Postnikov system of the mapping $f$ (it is also called the homotopic resolution, the $P$-system of general type or the Moore–Postnikov system of the mapping). For a constant mapping $c : X \to \text{pt}$ of a path-connected space $X$ its Postnikov system coincides with the Postnikov system of the space $X$.

In applications the so-called standard Postnikov systems (often called just Postnikov systems) are widely spread. These systems are Postnikov systems which consist of the principal fibrations $p_n: X_n \to X_{n-1}$ induced from the standard Serre fibrations $K(\pi_n, n) \to EK(\pi_n, n+1) \to K(\pi_n, n+1)$ by the Postnikov factors $k_n \in H^{n+1}(X_{n-1}; \pi_n)$ interpreted as mappings $k_n: X_{n-1} \to K(\pi_n, n+1)$ by virtue of the representation of the cohomology group as $H^{n+1}(X_{n-1}; \pi_n) \cong [X_{n-1}, K(\pi_n, n+1)]$. All spaces which are homotopy simple in all dimensions (Abelian spaces in the terminology of [2]) and only these spaces have standard Postnikov systems (see [3], [4]).

Standard Postnikov systems are applied to solve extension and lifting problems to which numerous problems in algebraic topology are reduced. The combined formulation of these problems is the following. Let a (homotopy) commutative square of spaces and mappings be given in which the mapping $i$ is a closed cofibration with cofibre $X/A$ and $p$ is a fibration with fibre $F$. The question is whether there exists a mapping $X\to Y$ such that both triangles obtained are (homotopy) commutative.

\begin{array}{ccc} A & \too & Y\\ i\big \downarrow & & \big \downarrow p\\ X & \too & B \end{array}

Further, if such a mapping does exist, then one is expected to determine the set $[X, Y]_B^A$ of homotopy classes of mappings $X \to Y$ "below A" (i.e. $\text{re } A$) and "above B" . Suppose that for the fibration $p : Y \to B$ there exists a standard Postnikov system $\{p_n: Y_n \to Y_{n-1},\, Y_0 = B\}$ (for this purpose, for example, it is sufficient for the spaces $Y$ and $B$ to be simply connected). The problem of relative lifting is solved step by step.

Consider the "elementary" problem of the relative lifting of a mapping $f_{n-1}: X \to Y_{n-1}$ from the $(n-1)$-st term of the Postnikov system to its $n$-th term:

\begin{array}{ccccc} A & \xrightarrow{g_{n-1}} & Y_n & \too & EK(\pi_n(F), n+1)\\ i \big \downarrow & & \big\downarrow p_n & & \big\downarrow\\ X & \xrightarrow[f_{n-1}]{} & Y_{n-1} & \xrightarrow[k_n]{} & K(\pi_n(F), n+1) \end{array}

The mappings $f_{n-1}$ and $g_{n-1}$ define a mapping $X/A \to K(\pi_n(F), n+1)$, i.e. a cohomology class $c^{n+1} \in H^{n+1}(X, A, \pi_n(F))$, called an obstruction. The mapping $f_{n-1}$ can be lifted to $Y_n$ if and only if $c^{n+1} = 0$. Two liftings $f_n'$ and $f_n''$ determine an element $d^n \in H^n(X, A; \pi_n(F))$, called a difference, which is equal to zero if and only if the liftings $f_n'$ and $f_n''$ are homotopic.

Thus, the problem of relative lifting is solved if the sequentially occurring obstructions $c^{n+1}$ vanish (for example, if $H^{n+1}(X, A; \pi_n(F)) = 0$). A lifting is unique if the sequentially occurring differences $d^n$ vanish (for example, if $H^n(X, A; \pi_n(F)) = 0$). In the case when the cofibration is an imbedding of CW-complexes, the obstruction $c^{n+1}$ and the difference $d^n$ coincide with usual "cell-wise" obstruction and difference (cf. Difference cochain and chain).

For simply-connected spaces $X$ with finitely-generated homology groups the Postnikov system is effectively computable [5] and, hence, the homotopy type of $X$ is effectively computable as well. However, in practice, for the majority of spaces one succeeds to compute only initial segments of Postnikov systems, which is due to the sharply increasing complexity of the computations. For computations one uses the method of cohomology operations (cf. Cohomology operation).

The dual of the Postnikov system is the Cartan–Serre system

$$\cdots \to X_n^{CS} \to X_{n-1}^{CS} \to \cdots \to X_{k-1}^{CS} = X$$

of a space $X$, consisting of fibrations whose fibres are the Eilenberg–MacLane spaces $K(\pi_n(X), n-1)$. The space $X_n^{CS}$ is called the $(n+1)$-st killing space for $X$. The terms $X_n^{CS}$ of the Cartan–Serre system are homotopy fibres of ($n-1$)-equivalences $\rho_n : X \to X_n$ for the Postnikov system of $X$, and the terms $X_n$ of a Postnikov system are loop spaces over the fibres of $X_n^{CS} \to X$.

A split Postnikov system is a sequence of principal fibrations

$$\cdots \to X_n \to X_{n-1} \to \cdots \to X_0 = \text{pt}$$ whose fibres are the Eilenberg–MacLane spaces $K(\pi_n, s_n)$, $s_n \le s_{n+1}$. Split Postnikov systems are the principal tool in the studies of so-called nilpotent spaces and, in particular, of their localizations (see Localization in categories, [2], [6], [7]). There also exist other versions of Postnikov systems (see [6]).

References

[1] M.M. Postnikov, "Studies on the homotopy theory of continuous mappings" , 1–2 , Moscow (1955) (In Russian)
[2] M.M. Postnikov, "Localization of topological spaces" Russian Math. Surveys , 32 : 6 (1977) pp. 121–184 Uspekhi Mat. Nauk , 32 : 6 (1977) pp. 117–181
[3] R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968) pp. Chapt. 13
[4] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[5] E.H. Brown, "Finite computability of Postnikov complexes" Ann. of Math. (2) , 65 (1957) pp. 1–20
[6] H.J. Baues, "Obstruction theory of homotopy classification of maps" , Springer (1977)
[7] P. Hilton, G. Mislin, J. Roitberg, "Localization of nilpotent groups and spaces" , North-Holland (1975)


Comments

References

[a1] G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. Chapt. IX
[a2] B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. Chapt. 17
How to Cite This Entry:
Postnikov system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Postnikov_system&oldid=17272
This article was adapted from an original article by S.N. Malygin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article