Given a Serre fibration between manifolds, how ugly can it be? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:44:52Z http://mathoverflow.net/feeds/question/116231 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116231/given-a-serre-fibration-between-manifolds-how-ugly-can-it-be Given a Serre fibration between manifolds, how ugly can it be? David Roberts 2012-12-13T03:21:05Z 2012-12-13T03:31:55Z <p>A Serre fibration is clearly defined with motivation from homotopy theory, but we can consider smooth versions $f\colon M\to N$ in the category of (finite-dimensional, paracompact etc) smooth manifolds, namely any smooth homotopy $H\colon I^n\times I \to N$ and smooth map $I^n\times\lbrace0\rbrace \to M$ lifting $H(-,0)$, there is a smooth lift of $H$ to $M$. For $N$ connected and $M$ non-empty, it is necessarily surjective. </p> <p>Clearly smooth fibre bundles are Serre fibrations, but arbitrary surjective submersions are not (for example: $[0,\frac{2}{3})\coprod (\frac{1}{3},1] \to [0,1]$). However, I have reasons to consider fibrations with connected fibre.</p> <p>I have a rough sketch of a proof in my head that smooth Serre fibrations are not far off being submersions, but that may just be the result of an overactive imagination.</p> <p>I'd like to know what's the ugliest that a smooth Serre fibration with connected fibres can be. Single examples are welcome, as are characterisation theorem.</p> <p>Secondly, what sort of fibres do we get? (Clearly they just might not be manifolds)</p> <p>Conversely, what nice properties are smooth Serre fibrations with connected fibres guaranteed to have?</p>