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Semen Podkorytov
Semen Podkorytov
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Let $p\colon E\to B$ be a fibration with fibres simply connected and homotopy equivalent to a compact CW-complex. Must $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ be surjective?

COMMENTS. Yes if $\pi_1(B)$ acts trivially on(EDIT) $H_*(\mathrm{fibre};\mathbb{Q})$$B$ is simply connected, even if the fibre is not compact but just finite-dimensional. In general, finite-dimensionality is not enough: consider the homotopy fibre sequence $\mathbb{R}^3\setminus\mathbb{Z}^3\to T^3\setminus\mathrm{point}\to T^3$.

MOTIVATION. If $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ is surjective, then any bundle gerbe over $p\colon E\to B$ is rationally trivial, cf. M. Murray, D. Stevenson, A note on bundle gerbes and infinite-dimensionality (http://arxiv.org/abs/1007.4922).

Let $p\colon E\to B$ be a fibration with fibres simply connected and homotopy equivalent to a compact CW-complex. Must $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ be surjective?

COMMENTS. Yes if $\pi_1(B)$ acts trivially on $H_*(\mathrm{fibre};\mathbb{Q})$, even if the fibre is not compact but just finite-dimensional. In general, finite-dimensionality is not enough: consider the homotopy fibre sequence $\mathbb{R}^3\setminus\mathbb{Z}^3\to T^3\setminus\mathrm{point}\to T^3$.

MOTIVATION. If $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ is surjective, then any bundle gerbe over $p\colon E\to B$ is rationally trivial, cf. M. Murray, D. Stevenson, A note on bundle gerbes and infinite-dimensionality (http://arxiv.org/abs/1007.4922).

Let $p\colon E\to B$ be a fibration with fibres simply connected and homotopy equivalent to a compact CW-complex. Must $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ be surjective?

COMMENTS. Yes if (EDIT) $B$ is simply connected, even if the fibre is not compact but just finite-dimensional. In general, finite-dimensionality is not enough: consider the homotopy fibre sequence $\mathbb{R}^3\setminus\mathbb{Z}^3\to T^3\setminus\mathrm{point}\to T^3$.

MOTIVATION. If $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ is surjective, then any bundle gerbe over $p\colon E\to B$ is rationally trivial, cf. M. Murray, D. Stevenson, A note on bundle gerbes and infinite-dimensionality (http://arxiv.org/abs/1007.4922).

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Semen Podkorytov
Semen Podkorytov
  • 809
  • 4
  • 11

Fibrations with non-simply connected base and rational homology

Let $p\colon E\to B$ be a fibration with fibres simply connected and homotopy equivalent to a compact CW-complex. Must $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ be surjective?

COMMENTS. Yes if $\pi_1(B)$ acts trivially on $H_*(\mathrm{fibre};\mathbb{Q})$, even if the fibre is not compact but just finite-dimensional. In general, finite-dimensionality is not enough: consider the homotopy fibre sequence $\mathbb{R}^3\setminus\mathbb{Z}^3\to T^3\setminus\mathrm{point}\to T^3$.

MOTIVATION. If $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ is surjective, then any bundle gerbe over $p\colon E\to B$ is rationally trivial, cf. M. Murray, D. Stevenson, A note on bundle gerbes and infinite-dimensionality (http://arxiv.org/abs/1007.4922).