SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS
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2019-01-01
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© 2019 The Author(s). We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to S2 but do not admit a spine (that is, a piecewise linear embedding of S2 that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer d invariants.
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Levine, AS, and T Lidman (2019). SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS. Forum of Mathematics, Sigma, 7. 10.1017/fms.2019.11 Retrieved from https://hdl.handle.net/10161/18625.
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Scholars@Duke
Adam S. Levine
My research is in low-dimensional topology, the study of the shapes of 3- and 4-dimensional spaces (manifolds) and of curves and surfaces contained therein. Classifying smooth 4-dimensional manifolds, in particular, has been a deep challenge for topologists for many decades; unlike in higher dimensions, there is not enough "wiggle room" to turn topological problems into purely algebraic ones. Many of my projects reveal new complications in the topology of 4-manifolds, particularly related to embedded surfaces. My main tools come from Heegaard Floer homology, a powerful package of invariants derived from symplectic geometry. I am also interested in the interrelations between different invariants of knots in 3-space, particularly the connections between knot invariants arising from gauge theory and symplectic geometry and those coming from representation theory.
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