5
$\begingroup$

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 $\pi_1$.

$\endgroup$
2
  • 1
    $\begingroup$ What is the relationship of your tower with the Moore-Postnikov tower? $\endgroup$
    – Mark Grant
    Jul 9, 2017 at 14:24
  • $\begingroup$ @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. $\endgroup$
    – res
    Jul 10, 2017 at 10:57

1 Answer 1

3
$\begingroup$

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"

$\endgroup$
5
  • $\begingroup$ 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 $X\xrightarrow{f} Y$ $\endgroup$
    – res
    Jul 10, 2017 at 11:10
  • $\begingroup$ 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 $?\to F_i$ $\endgroup$
    – res
    Jul 10, 2017 at 13:58
  • $\begingroup$ @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 $Y$. $\endgroup$
    – John Klein
    Jul 10, 2017 at 14:29
  • $\begingroup$ 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. $\endgroup$
    – res
    Jul 10, 2017 at 15:01
  • 1
    $\begingroup$ @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) $\endgroup$
    – John Klein
    Jul 10, 2017 at 20:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.