Postnikov system

In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type. For a space X , this is a list of spaces \{X_n\}_{n \geq 0} where \pi_k(X_n) = \begin{cases} \pi_k(X) & \text{ for } k \leq n \\ 0 & \text{ for } k > n \end{cases} and a series of maps \phi_n: X_n \to X_{n-1} that are fibrations with Eilenberg-MacLane spaces K(\pi_n(X),n) as fibers. In short, we are decomposing the homotopy type of X using an inverse system of topological spaces whose homotopy type at degree k agrees with the truncated homotopy type of the original space X . Postnikov systems were introduced by, and are named after, Mikhail Postnikov. There is a similar construction called the Whitehead tower (defined below) where instead of having spaces X_n with the homotopy type of X for degrees \leq n , these spaces have null homotopy groups \pi_{k}(X_n) =0 for 1 .

In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type. For a space X , this is a list of spaces \{X_n\}_{n \geq 0} where \pi_k(X_n) = \begin{cases} \pi_k(X) & \text{ for } k \leq n \\ 0 & \text{ for } k > n \end{cases} and a series of maps \phi_n: X_n \to X_{n-1} that are fibrations with Eilenberg-MacLane spaces K(\pi_n(X),n) as fibers. In short, we are decomposing the homotopy type of X using an inverse system of topological spaces whose homotopy type at degree k agrees with the truncated homotopy type of the original space X . Postnikov systems were introduced by, and are named after, Mikhail Postnikov. There is a similar construction called the Whitehead tower (defined below) where instead of having spaces X_n with the homotopy type of X for degrees \leq n , these spaces have null homotopy groups \pi_{k}(X_n) =0 for 1 . \pi_i(X_n) = 0 for i > n . Each map p_n : 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) . The first two conditions imply that 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 are contractible. Note the third condition is only included optionally by some authors. giving a homotopy type with two non-trivial homotopy groups, \pi_1(X) = G , and \pi_n(X) = A . Then, from the previous discussion, the fibration map BG \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. \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(\Z,2) , and X_3 comes from a pullback sequence \begin{matrix} X_3 & \to & * \\ \downarrow & & \downarrow \\ X_2 & \to & K(\Z,4) , \end{matrix} which is an element in [p_3] \in [K(\Z,2), K(\Z,4)] \cong H^4(\mathbb{CP}^\infty) = \Z . If this was trivial it would imply X_3 \simeq K(\Z,2)\times K(\Z,3) . But, this is not the case! In fact, this is responsible for why strict infinity groupoids don't model homotopy types. Computing this invariant requires more work, but can be explicitly found. This is the quadratic form x \mapsto x^2 on \Z \to \Z coming from the Hopf fibration S^3 \to S^2 . Note that each element in H^4(\mathbb{CP}^\infty) gives a different homotopy 3-type. E^2_{p,q} = H_p\left(K(\Z, n), H_q\left(K\left(\pi_{n+1}\left(S^n\right), n + 1\right)\right)\right) . And the first non-trivial map to \pi_{n+1}\left(S^n\right) , d^{n+1}_{0,n+1} : H_{n+2}(K(\Z, n)) \to H_0\left(K(\Z, n), H_{n+1}\left(K\left(\pi_{n+1}\left(S^n\right), n + 1\right)\right)\right) , equivalently written as d^{n+1}_{0,n+1} : H_{n+2}(K(\Z, n)) \to \pi_{n+1}\left(S^n\right) . If it's easy to compute H_{n+1}\left(S^n_{n+1}\right) and H_{n+2}\left(S^n_{n+2}\right) , then we can get information about what this map looks like. In particular, if it's an isomorphism, we obtain a computation of \pi_{n+1}\left(S^n\right) . For the case n = 3 , this can be computed explicitly using the path fibration for K(\Z, 3) , the main property of the Postnikov tower for \mathfrak{X}_4 \simeq S^3 \cup \{\text{cells of dimension} \geq 6\} (giving H_4(X_4) = H_5(X_4) = 0 , and the universal coefficient theorem giving \pi_4\left(S^3\right) = \Z/2 . Moreover, because of the Freudenthal suspension theorem this actually gives the stable homotopy group \pi_1^\mathbb{S} since \pi_{n+k}\left(S^n\right) is stable for n \geq k + 2 . Note that similar techniques can be applied using the Whitehead tower (below) for computing \pi_4\left(S^3\right) and \pi_5\left(S^3\right) , giving the first two non-trivial stable homotopy groups of spheres. \pi_i^{\mathbb{S}}\left(E_{(n)}\right) = 0 for i > n , \left(\tau_n\right)_* : \pi_i^{\mathbb{S}}(E) \to \pi_i^{\mathbb{S}}\left(E_{(n)}\right) is an isomorphism for i \leq n , where \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. The lower homotopy groups are zero, so \pi_i(X_n) = 0 for i \leq n . The induced map \pi_i : \pi_i(X_n) \to \pi_i(X) is an isomorphism for i > n . The maps X_n \to X_{n-1} are fibrations with fiber K(\pi_n(X), n-1) . 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 p , then the induced map X_{n+1} \to X_n is a fiber bundle with fiber homeomorphic to \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 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 \pi_i(X_n) = \pi_i\left(X_{n-1}\right) for i \geq n + 1 , \pi_i(X_n) = \pi_i(X_{n-1}) = 0 for i , and finally, there is an exact sequence 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 X_n \to K(\pi_{n+1}(X), n + 1) and noting that the Eilenberg–Maclane space has a cellular decomposition X_{n-1} \cup \{\text{cells of dimension} \geq n + 2\} ; thus, \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. \pi_k(X^n) = \begin{cases} \pi_k(X) & k > n \\ 0 & k \leq n \end{cases} E\langle n \rangle = \operatorname{Hofiber}\left(\tau_n: E \to E_{(n)}\right) then this can be organized in a tower giving connected covers of a spectrum. This is a widely used construction in bordism theory because the coverings of the unoriented cobordism spectrum M\text{O} gives other bordism theories \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.

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