I've found myself in the following situation and the tools that have been suggested to me to calculate the necessary invariants don't quite seem up to the job (I may be wrong in this as I have very little experience with spectral sequences - something I'm working actively to resolve).

I have a space $\Omega$ which fibers over the circle via a map $p$. The space $\Omega$ is not a traditionally 'nice' space, as the fiber $F$ of the bundle $p$ is a Cantor set, and so $\Omega$ is in particular not a path connected space or a $CW$-complex. It is connected however.

Associated to $\Omega$ are two similar space $\Omega_1$ and $\Omega_2$ (they both fiber over the circle by maps $p_1$ and $p_2$ respectively) with the following property.

The product bundle $\Omega_1\times\Omega_2\stackrel{(p_1, p_2)}{\rightarrow} S^1\times S^1$, together with the diagonal inclusion $S^1\stackrel{\Delta}\rightarrow S^1\times S^1$, have as their topological pullback $\Omega$ with $p\colon\Omega\rightarrow S^1$ and an inclusion map $i\colon\Omega\rightarrow \Omega_1\times\Omega_2$.

Diagramatically, the following diagram commutes and satisfies the usual universal property of pullbacks:

$$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} \Omega & \ra{i} & \Omega_1\times\Omega_2 \\ \da{p} & & \da{(p_1,p_2)} \\ S^1 & \ras{\Delta} & T^2 \\ \end{array} $$

Now, in principal I know the cohomology of $\Omega_1$ and $\Omega_2$ over field coefficients (say $\mathbb{Q}$), and so their product via a Kunneth formula, and can calculate the induced map $H^n(T^2)\rightarrow H^n(\Omega_1\times\Omega_2)$. Given this information, I wonder if it is possible to calculate the cohomology groups of $\Omega$.

As far as I understand, this situation isn't uncommon and, in the case of a simply connected base space in the right bundle, this can be tackled using the Eilenberg-Moore spectral sequence (apologies but my current knowledge of spectral sequences is limited so I am not sure how correct this is). Is there a generalised method which can be applied to base spaces which are not simply connected? Is a spectral sequence approach the best/only approach to calculating the cohomology of $\Omega$ given this information?

  • $\begingroup$ There is an Eilenberg-Moore spectral sequence for general connected base spaces; the trouble is that it doesn't always converge to thing you want unless the action of $\pi_1$ satisfies some nilpotent conditions. I can only find a reference for the case of a fibration, but the case of a homotopy pullback shouldn't be too different: www3.nd.edu/~wgd/Dvi/StrongConvergenceEilenbergMoore.pdf $\endgroup$ – Dylan Wilson Jul 29 '13 at 14:25

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