The trace embedding lemma and spinelessness

Abstract

We demonstrate new applications of the trace embedding lemma to the study of piecewiselinear 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 S 2 which have smooth structures distinguished by several formal properties: W is diffeomorphic to a knot trace but W is not, W contains S 2 as a smooth spine but W does not even contain S 2 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.

3-manifolds that bound no definite 4-manifold

“…In Section 4 we will give a more direct proof that the 3-manifold Y is not obtained as Dehn surgery along a knot in S 3 , and use Y to produce another example of a spineless 4-manifold (see [13] and [8] for previous work on the subject-Hayden and Piccirillo's results in particular are much stronger than ours).…”

Traces of links and simply connected 4‐manifolds

“…Inspired by the higher genus traces defined by Hayden and Piccirillo in [7], we generalize this concept to the link setting in the way that it can be applied to give an alternative characterization of the slice genus. We start by describing the two main constructions, leaving the general case for later: The first 4-manifold, which here we denote with 𝑋 g,1 (𝐾), is the genus g 2-handles attached along the knot 𝐾 in 𝑆 3 (with framing 0) appearing in [7]; while the second one consists of attaching a planar (genus zero) 2-handles with 𝓁 boundary component along an 𝓁-component link in 𝑆 3 , and we call it 𝑋 0,𝓁 (𝐿).…”

“…This naturally extends the usual genus function of a closed 4-manifold to the setting of 4-manifolds with boundary, and this concept can be interesting even in the case when the manifold at hand has no second homology. This approach requires the extension of the trace embedding lemma to be relevant in this context; in particular, we will attach higher genus handles (as it has been already considered in [7]) along framed knots and links, and will consider higher order traces (when the core surfaces of the handles have potentially multiple boundary components).…”

Traces of links and simply connected 4-manifolds

“…Inspired by the higher genus traces defined by Hayden and Piccirillo in [7], we generalize this concept to the link setting in the way that it can be applied to give an alternative characterization of the slice genus. We start by describing the two main constructions, leaving the general case for later: the first 4-manifold, which here we denote with X g,1 0 (K), is the genus g 2-handle attached along the knot K in S 3 (with framing 0) appearing in [7]; while the second one consists of attaching a planar (genus zero) 2-handle with boundary component along an -component link in S 3 , and we call it X 0, 0 (L).…”

“…Obviously, in order to specify the diffeomorphism f we also need to fix a 0 K g times Figure 2: A Kirby diagram of X g,1 0 (K). The picture is taken from [7].…”