Embedding Heegaard Decompositions

A smooth embedding of a closed $3$-manifold $M$ in $\mathbb{R}^4$ may generically be composed with projection to the fourth coordinate to determine a Morse function on $M$ and hence a Heegaard splitting $M=X\cup_\Sigma Y$. However, starting with a Heegaard splitting, we find an obstruction coming from the geometry of the curve complex $C(\Sigma)$ to realizing a corresponding embedding $M\hookrightarrow \mathbb{R}^4$.

A Fox–Milnor theorem for knots in a thickened surface

A knot in a thickened surface [Formula: see text] is a smooth embedding [Formula: see text], where [Formula: see text] is a closed, connected, orientable surface. There is a bijective correspondence between knots in [Formula: see text] and knots in [Formula: see text], so one can view the study of knots in thickened surfaces as an extension of classical knot theory. An immediate question is if other classical definitions, concepts, and results extend or generalize to the study of knots in a thickened surface. One such famous result is the Fox–Milnor Theorem, which relates the Alexander polynomials of concordant knots. We prove a Fox–Milnor Theorem for concordant knots in a thickened surface by using Milnor torsion.

EMBEDDINGS, NORMAL INVARIANTS AND FUNCTOR CALCULUS

This paper investigates the space of codimension zero embeddings of a Poincaré duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincaré immersions to a certain space of “unlinked” Poincaré embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie’s homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.

On smoothness of the Banach space embedding

For a Banach space X, smoothness at a point of the natural embedding ◯ in X**, is characterised by a continuity property of the support mapping from X into X*. It then becomes clear that there are many non-reflexive Banach spaces with smooth embedding, a matter of interest raised by Ivan Singer [Bull. Austral. Math. Soc. 12 (1975), 407–416].

On an unlinking theorem

An n-link of multiplicity is a smooth embedding of the disjoint union of μ copies of Sn in Sn+2; is said to be trivial if it extends to a smooth embedding of the disjoint union of μ copies of Dn+1. Let , and Cnμ denote the wedge product of μ copies of S1 and (μ – 1) copies of Sn+1. Then clearly, if is trivial, then X ≃ Cn,μ, where ≃ denotes homotopy equivalence.