2019/12/30 by Kyle Hayden, Hayden, Kyle, Lisa Piccirillo +1 · 2 citations
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1912.13021
We demonstrate new applications of the trace embedding lemma to the study of piecewise-linear surfaces and the detection of exotic phenomena in dimension four. We provide infinitely many pairs of homeomorphic 4-manifolds W and W' homotopy equivalent to S2 which have smooth structures distinguished by several formal properties: W' is diffeomorphic to a knot trace but W is not, W' contains S2 as a smooth spine but W does not even contain S2 as a piecewise-linear spine, W' is geometrically simply connected but W is not, and W' does not admit a Stein structure but W does. In particular, the simple spineless 4-manifolds W provide an alternative to Levine and Lidman's recent solution to Problem 4.25 in Kirby's list. We also show that all smooth 4-manifolds contain topological locally flat surfaces that cannot be approximated by piecewise-linear surfaces.