Ramified cover of 4-sphere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T20:59:10Z http://mathoverflow.net/feeds/question/8697 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8697/ramified-cover-of-4-sphere Ramified cover of 4-sphere Anton Petrunin 2009-12-12T17:49:58Z 2010-04-23T16:04:06Z <p>Is it true that any closed oriented \$4\$-dimensional manifold can be obtained as a result of the following construction: </p> <p>Take \$S^4\$ with a finite collection of immersed closed 2-manifolds (with transversal intersections and self-intersections) and construct ramified cover of \$S^4\$ with a ramification of order at most 2 only at these submanifolds.</p> <p><strong>Comments:</strong></p> <ul> <li><p>Two related questions: <a href="http://mathoverflow.net/questions/5546/ramified-covers-of-3-torus" rel="nofollow">Ramified covers of 3-torus</a>, <a href="http://mathoverflow.net/questions/5618/ramified-covers-of-sn" rel="nofollow">Ramified covers of \$S^n\$</a></p></li> <li><p>According to Feighn's <a href="http://dmle.cindoc.csic.es/pdf/COLLECTANEAMATHEMATICA%5F1986%5F37%5F01%5F04.pdf" rel="nofollow">Branched covers according to J.W. Alexander</a> any closed oriented 4-manifold is a branched cover of \$S^4\$ with a ramification along 2-skeleton of 4-tetrahedron embedded in \$S^4\$ (which is not at all a 2-manifold).</p></li> </ul> http://mathoverflow.net/questions/8697/ramified-cover-of-4-sphere/8715#8715 Answer by Allan Edmonds for Ramified cover of 4-sphere Allan Edmonds 2009-12-12T22:39:41Z 2009-12-12T22:39:41Z <p>The answer is yes, at least if we interpret your phrase "ramification of order 2" to mean "simple branched covering". See Piergallini, R., Four-manifolds as \$4\$-fold branched covers of \$S^4\$. Topology 34 (1995), no. 3, 497--508. Any closed, orientable PL 4-manifold can be expressed as a 4-fold simple branched covering of S<sup>4</sup> branched along an immersed surface with only transverse double points. It is apparently still an open question whether the branch set can be chosen to be nonsingular. A simple branched covering of degree d is a branched covering in which each branch point is covered by d-1 points, only one of which is singular, of local degree 2.</p>