Surface slices and homology spheres

TL;DR

The paper addresses which 3-manifolds embed in $S^4$ versus in homology 4-spheres, showing there exist infinitely many integer homology 3-spheres embeddable in a homology 4-sphere but not in any homotopy 4-sphere. It develops a Taubes-based gauge-theoretic obstruction and a diagrammatic surface-slicing framework using spun knots and branched covers to realize these obstructions. The main contributions include constructing definite cobordisms from even symmetric unions, a fundamental-group criterion via pushout diagrams, and a hyperbolic-volume argument to produce infinitely many distinct embeddable manifolds. This work clarifies the boundary between embeddability in $S^4$ and in homology 4-spheres, providing new smooth-category obstructions and constructive examples through a combination of gauge theory, knot theory, and 4-manifold topology.

Abstract

We develop the theory of the diagrammatics of surface cross sections to prove that there are an infinite number of homology 3-spheres smoothly embeddable in a homology 4-sphere but not in a homotopy 4-sphere. Our primary obstruction comes from work of Taubes.

Figures (10)

• Figure 1: An example of a knot whose double branched cover embeds in a homology sphere but not a homotopy sphere.
• Figure 2: The above figure illustrates the coordinate pushing in Proposition \ref{['prop:combin']} restricted to an example double arc. In black we have the arc, with the colored arcs above and below representing the two preimages. The points in each preimage are mapped to the black arc via projection to the x coordinate. In red are the intersections of $\gamma$ with the double arc on the above arc and below arc respectively, and the green and blue components of the colored arcs represent $\mathcal{F}_1$ and $\mathcal{F}_2$ respectively. On the right picture we have an embedded version of the picture given by pushing the blue arcs up in the projected coordinate and pushing the green arcs down in the projected coordinate, fixing $\gamma$. We note that the resolved copy is not only embedded, but the prescribed over sheet is above the under sheet.
• Figure 3: The tangle of the trefoil knot, its corresponding chord diagram (with the arrows pointing to the over crossing), and the decker set of the corresponding spun knot broken surface diagram projected so that latitudinal circles are mapped to horizontal lines. The dotted lines correspond to the lower sheets, with the solid lines being the upper sheets. The $\alpha_i$ and $\beta_i$ for the pairs of curves match up each point in a given latitude line with its corresponding point in the same longitudinal line, and the curves are matched up according to the chord diagram.
• Figure 4: $K\hash \overline{K}$ drawn on the decker set for the spun trefoil, and the corresponding knot diagram.
• Figure 5: An even symmetric union over the trefoil drawn on the decker set for the spun trefoil, and the corresponding knot diagram in green. The blue arcs show the discs in the canonical ribbon disc for the symmetric union.

Theorems & Definitions (18)

• Theorem 3
• Theorem 4
• Corollary 5
• proof : Idea of proof
• Theorem 6
• proof
• Definition 7
• Proposition 8
• proof
• Example 9