# Delooping a fibration sequence with loopspace fiber and finite CW complexes

The following question is somewhat similar to a previous one on MathOverflow, except that my application does not directly involve Eilenberg-MacLane spaces $$K(G,n)$$, and so I don't see the immediate need for $$n$$-simple maps as in the theory of Postnikov $$k$$-invariants.

Suppose that $$\Omega T \longrightarrow F \longrightarrow E$$ is a homotopy fibration sequence, where $$T$$ and $$F$$ are connected finite CW complexes. Is there a quantification in some sort of obstruction theory for extending this sequence further to the right in the form $$F \longrightarrow E \longrightarrow T$$ ? This is a 1950s-era topology question.

Of course, the original sequence extends further to the right as $$F \longrightarrow E \longrightarrow B hAut(\Omega T)$$. First, this is by Stasheff's paper for finite CW complexes as the fiber and any CW base, working with products in the full topological category [Sta63]. Later, this is by May's simplicial upgrade to infinite CW complexes as the fiber, working in the compactly generated category [9.5, 9.8][May75].

In my application, $$T$$ happens to have the structure of a Lie group, but this is a happy accident of its dimension, and I'm hoping for an answer that can work independent of this fact.

[Sta63]: James D Stasheff, A classification theorem for fibre spaces, Topology 2, 1963

[May75]: J Peter May, Classifying spaces and fibrations, Memoirs AMS 1:155, 1975

EDIT: As suggested, I renamed the base space $$B$$ to $$T$$, in order to not confuse it with the classifying space functor. Also, I added two citations.

• (I rename $B$ to $T$ to avoid confusion) I think the only obstruction is factorizing action map $E \to BAut(\Omega T)$ through delooping of tautological action $T \to BAut(\Omega T)$ which is done by usual obstruction theory. – Denis T. Oct 22 '19 at 21:50
• @Denis T. : Любезно, what is the explicit formula for this tautological action? Does it it assume that $T$ is a topological group, say by pointwise-conjugating a loop by an element of $T$? If so, isn't this instead a map $T \longrightarrow Aut(\Omega T)$? – Qayum Khan Oct 22 '19 at 22:49
• @QayumKhan The tautological action is the one corresponding to the pathspace fibration $\Omega T\to P T\to T$ (morally it is the action of $\Omega T$ on itself by left(?) multiplication) – Denis Nardin Oct 23 '19 at 7:48
• @QayumKhan DenisNardin is right, I mean just an application of classifying space functor to map $\Omega T \to Aut(\Omega T)$ representing left (more precisely, the side $\pi_1$ of base acts on fiber in your preferrable conventions) multiplication. Also possibly you want to use Moore loops for that to avoid some nuances with non-strictly associative actions etc. – Denis T. Oct 23 '19 at 16:55

This question is addressed in the paper

Ganea, T., Induced fibrations and cofibrations, Trans. Am. Math. Soc. 127, 442-459 (1967). ZBL0149.40901.

A first observation is that $$\Omega T\to F\to E$$ extends to the right if and only if it is induced from the based path fibration $$\Omega T\to PT\to T$$ by a map $$p:E\to T$$. In Section 2, various sufficient conditions for such a fibration to be induced are given.

Two sample results:

Corollary 2.5: Suppose that $$\pi_q(E)\neq 0$$ only if $$m\le q\le n + m-1$$ and that $$\pi_q(\Omega T) \neq 0$$ only if $$n\le q\le n + m-1$$, where $$n\ge m\ge 2$$. If the Whitehead product pairing $$W:\pi_n(\Omega T) \otimes \pi_m(F)\to \pi_{n + m-1}(F)$$ vanishes, then $$\Omega T\to F\to E$$ is induced.

Theorem 2.10: Suppose $$\Omega T$$, $$F$$ and $$E$$ all have the homotopy type of aspherical CW complexes. Then $$\Omega T\to F\to E$$ is induced if and only if the image of the induced map $$\pi_1(\Omega T)\to \pi_1(F)$$ lies in the center.

• I would be very interested in knowing the level of generality in which these results hold. Do they work, for instance, in any $\infty$-topos? – skd Oct 28 '19 at 0:26
• @MarkGrant : I appreciate the partial answer; however, my specific application (which I didn't specify in the question) fails to satisfy the connectivity hypotheses of your answer here and also the previously linked MathOverflow question. Specifically, all of my finite CW complexes $T, F, E$ are non-simply connected and have homotopy groups in infinitely many dimensions. My opinion is that the results in Ganea's paper are more-or-less a 'thickening' of the preexisting result on Eilenberg--MacLane spaces that he advances. – Qayum Khan Oct 28 '19 at 17:35