THE TRACE EMBEDDING LEMMA AND SPINELESSNESS

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.