Postnikov system

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)

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