# Postnikov-type tower for a map between spaces

Part I

Let $f: X\to Y$ be an arbitrary map between topological spaces/simplicial sets $X$ and $Y$. Then we can associate with $f$ a tower of fibrations $$\dots \to F_i\to\dots \to F_1\to F_0=Y$$

constructed in a following way:

0.let $F_0 = Y$ and we take $C_0$ as homotopy cofiber of $f$

1.define $F_1$ as homotopy fiber of $Y\to C_0$ and since composition $X\to Y\to C_0$ is null-homotopic we have a map $X\to F_1$. $C_1$ is a homotopy cofiber of this map.

2.$F_2$ is a homotopy fiber of $Y\to C_1$ and so on.

Therefore we got a tower of principle (under some assumptions on $\pi_1$-action I believe) fibrations $$\Omega C_i\to F_{i+1}\to F_i$$ together with collection of maps $X\to F_i$. Many questions about this construction can be asked. What does this tower tell us about $f$? About $X$ ? Under what conditions on $f$ $X\to \lim F_i$ is an equivalence ? etc

Q1: I'm sure such construction have been studied before, where I can find any information about it?

Part II

Actually, Postnikov tower for a space $X$ is constructed in a very similar way, with one more additional step: instead of taking homotopy fiber straight away, we kill homotopy groups of cofiber using fundamental class.

In particular, we start with $X\to K(\pi_1 X,1)$, take its homotopy cofiber $C_0$ and define next step of a tower as a homotopy fiber of composition $K(\pi_1 X, 1)\to C_0\to K(\pi_3 C_0, 3)$. Here $C_0$ is $2$-connected and $\pi_3 C_0=\pi_2 X$.

Q2 What is the phylosophical'' meaning of taking this additional step $C_n\to K(\pi_{n+2} X,n+3)$ ? What will happen with Postnikov tower if we will not kill homotopy groups of cofiber, apart from losing nice description of $k$-invariants as cohomology classes ?

P.S. In general, I would like to compare two constructions (without killing homotopy groups of cofiber and with it) and to understand, which one is more canonical''. Conjecturally, taking different morphisms $f$ (for example $X\to \Omega^{\infty}\Sigma^{\infty} X$) we will get towers, that a related to already known constructions (such as Goodwillie tower, for example).

EDIT

Actually (if I'm not wrong), if $X$ is simple, we can start the Postnikov tower for $X$ straight away from a map $X\to *$. Fundamental class of $\Sigma X=\mathrm{hocofib\{X\to *\}}$ is then represented by $\Sigma X\to K(\pi_1 X, 2)$ and $X_1=\mathrm{hofib}\{*\to K(\pi_1 X,2)\}=K(\pi_1 X,1)$. If we apply my construction to a map $X\to *$, then as $F_1$ we will have $\Omega\Sigma X$, instead of $X_1=K(\pi_1 X,1)$ in a usual Postnikov tower.

In general, comparing two constructions, we have a map $F_i\to X_i$, in the example above in the level one this is $\Omega\Sigma X\to K(\pi_1 X,1)$, which is an isomorphism on $\pi_1$.

• What is the relationship of your tower with the Moore-Postnikov tower? – Mark Grant Jul 9 '17 at 14:24
• @MarkGrant I've checked Hatcher's book for Moore-Postnikov tower, looks like this is slightly different thing, I can't see the obvious connection. I'm thinking about my tower as a "resolution" of $X$, constructed from "representation" $X\xrightarrow{f} Y$, Moore-Postnikov tower looks more like approximation of $f$ itself. I will also edit my post a little bit to include connection with Postnikov tower. – res Jul 10 '17 at 10:57

Here is an answer to your "Part I."

The construction you outlined (and its dual) was considered in detail in the papers of Ganea. In particular:

Ganea, T. Induced fibrations and cofibrations. Trans. Amer. Math. Soc. 127 1967 442–459.

(See Section 3 of the above for your case.)

For the dual question see my answer to: Fibrations and Cofibrations of spectra are "the same"

• Thanks for the paper John, will check it out. Another interesting (but vague) question for me is how the dual version of this tower connected with a Cech resolution $\dots X\times_Y X\rightrightarrows X\xrightarrow{f} Y$ of a map $X\xrightarrow{f} Y$ – res Jul 10 '17 at 11:10
• Also I'm curious, what is the analog of "killing higher homotopy groups of cofiber" in the dual construction. Looks like we need to compose inclusions of homotopy fibers $F_i$ with some maps $?\to F_i$ – res Jul 10 '17 at 13:58
• @res As to your first comment, I don't think the resolutions are identical. The $n$-th term of the Cech resolutuion is the fiberwise space $X \ast_Y \cdots \ast_Y X$ (the $n$-fold fiberwise join of $X$ over $Y$), whereas the Ganea resolution has $n$-th term $X \ast_Y PY \ast_Y \cdots \ast_Y PY$, where $PY$ is the based path space of $Y$. – John Klein Jul 10 '17 at 14:29
• Agreed that they need not to be equivalent, it may be too strong. I will be happy just with a map between them. I need to think more about how to wrap up this Ganea tower in a simplicial space. About the paper, I couldn't find a nice close form for cofibers in my original construction, similar to $F\wedge (\Sigma\Omega Y)^{\wedge j}$ in the dual Ganea tower. Sad. – res Jul 10 '17 at 15:01
• @res There is a map from the Ganea resolution to the Cech resolution: we can assume $X\to Y$ is a fibration. Choose a basepoint in X, then the path fibration $PY \to Y$ admits a factorization $PY \to X\to Y$. There is then a map of fiberwise spaces $X \times_Y PY \times_Y\cdots \times_Y PY \to X \times_Y X \times_Y\cdots \times_Y X$ which is a morphism of simplicial spaces. This gives a map of resolutions (by taking skeleta of the realizations) – John Klein Jul 10 '17 at 20:50