Mapping torus of a homotopy equivalence - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T18:52:58Z http://mathoverflow.net/feeds/question/5239 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5239/mapping-torus-of-a-homotopy-equivalence Mapping torus of a homotopy equivalence RJR 2009-11-12T19:44:33Z 2011-12-19T07:42:02Z <p>The mapping torus $M_f$ of a homeomorphism $f$ of some topological space $X$ is a fiber bundle whose base is a circle and whose fiber is the original space $X$. If instead of a homeomorphism $f$ is just a homotopy equivalence of $X$, is $M_f$ a fibration over the circle with fiber homotopic to $X$? </p> http://mathoverflow.net/questions/5239/mapping-torus-of-a-homotopy-equivalence/5242#5242 Answer by Charles Rezk for Mapping torus of a homotopy equivalence Charles Rezk 2009-11-12T19:55:15Z 2009-11-12T19:55:15Z <p>There's no reason to expect $M_f$ to be a fiber bundle, or even a Hurewicz or Serre fibration. (Think of $f: \mathbb{R} \to \mathbb{R}$ by $f(x)=0$. This is a homotopy equivalence, but $M_f$ is certainly not locally trivial, nor does $M_f \to S^1$ have any nice lifting properties.)</p> <p>What <em>is</em> true is that the <em>homotopy fiber</em> of $M_f\to S^1$ is weakly equivalent to $X$, if $f$ is a homotopy equivalence (or even a weak equivalence). This often gets proved by using the theory of "quasi-fibrations". </p> http://mathoverflow.net/questions/5239/mapping-torus-of-a-homotopy-equivalence/5276#5276 Answer by Eric Wofsey for Mapping torus of a homotopy equivalence Eric Wofsey 2009-11-12T22:58:52Z 2009-11-12T22:58:52Z <p>This is closely related to <a href="http://mathoverflow.net/questions/2665/is-a-map-that-is-locally-fiberwise-equivalent-to-a-product-a-hurewicz-fibration" rel="nofollow">a question of mine</a>, which was motivated by wondering whether the mapping cylinder of a homotopy equivalence is a fibration over an interval. The counterexample given there (for the homotopy equivalence from I to a point) also gives a counterexample for the mapping torus, and makes it easy to see how it goes wrong.</p> http://mathoverflow.net/questions/5239/mapping-torus-of-a-homotopy-equivalence/83852#83852 Answer by Andrew Ranicki for Mapping torus of a homotopy equivalence Andrew Ranicki 2011-12-19T07:42:02Z 2011-12-19T07:42:02Z <p>Check out </p> <p>Homotopy equivalences and mapping torus projections D. S. Coram, P. F. Duvall Fund. Math. 109 (1980), 1-7</p> <p><a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm109/fm10911.pdf" rel="nofollow">http://matwbn.icm.edu.pl/ksiazki/fm/fm109/fm10911.pdf</a></p>