We study locally flat disks in $(\mathbb {C}P^2)^\circ :=({\mathbb {C}} P^2)\setminus \mathring {B}^4$ with boundary a fixed knot $K$ and whose complement has fundamental group $\mathbb {Z}$. We show that, up to topological isotopy relative to the (rel.) boundary, such disks necessarily arise by performing a positive crossing change on $K$ to an Alexander polynomial one knot and capping off with a $\mathbb {Z}$-disk in $D^4.$ Such a crossing change determines a loop in $S^3 \setminus K$ and we prove that the homology class of its lift to the infinite cyclic cover leads to a complete invariant of the disk. We prove that this determines a bijection between the set of rel. boundary topological isotopy classes of $\mathbb {Z}$-disks with boundary $K$ and a quotient of the set of unitary units of the ring $\mathbb {Z}[t^{\pm 1}]/(\Delta _K)$. Number-theoretic considerations allow us to deduce that a knot $K \subset S^3$ with quadratic Alexander polynomial bounds $0,1,2,4$, or infinitely many $\mathbb {Z}$-disks in $(\mathbb {C}P^2)^\circ$. This leads to the first examples of knots bounding infinitely many topologically distinct disks whose exteriors have the same fundamental group and equivariant intersection form. Finally, we give several examples where these disks are realized smoothly.

The trace of the $n$-framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$-dimensional knot invariants. For each $n$, this provides conditions that imply a knot is topologically $n$-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

A bilinear map $\varPhi :\mathbb {R}^r\times \mathbb {R}^s\to \mathbb {R}^n$ is nonsingular if $\varPhi (\overrightarrow {a},\overrightarrow {b})=\overrightarrow {0}$ implies $\overrightarrow {a}=\overrightarrow {0}$ or $\overrightarrow {b}=\overrightarrow {0}$. These maps are of interest to topologists, and are instrumental for the study of vector bundles over real projective spaces. The main purpose of this paper is to produce examples of such maps in the range $24\leqslant r\leqslant 32,\ 24\leqslant s\leqslant 32,$ using the arithmetic of octonions (otherwise known as Cayley numbers) as an effective tool. While previous constructions in lower dimensional cases use ad hoc techniques, our construction follows a systematic procedure and subsumes those techniques into a uniform perspective.

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.

For the mapping class group of 3-manifold with respect to a Heegaard splitting, a simplicial complex is constructed such that its group of automorphisms is identified with the mapping class group.

We prove that simple, thick hyperbolic P-manifolds of dimension at least three exhibit Mostow rigidity. We also prove a quasi-isometry rigidity result for the fundamental groups of simple, thick hyperbolic P-manifolds of dimension at least three. The key tool in the proof of these rigidity results is a strong form of the Jordan separation theorem, for maps from $S^n\rightarrow S^{n+1}$ which are not necessarily injective.

If $V\,\to \,X$ is a vector bundle of fiber dimension $k$ and $Y\,\to \,X$ is a finite sheeted covering map of degree $d$ , the implications for the Euler class $e(V)$ in ${{H}^{k}}(X)$ of $V$ implied by the existence of an embedding $Y\,\to \,V$ lifting the covering map are explored. In particular it is proved that $d{{d}^{\prime }}\text{e(V)}\text{=}\text{0}$ where ${{d}^{\prime }}$ is a certain divisor of $d\,-\,1$ , and often ${{d}^{\prime }}=1$ .

We classify the compact 3-manifolds whose boundary is a union of 2-spheres, and which embed in T ×I ×I, where T is a triod and I the unit interval. This class is described explicitly as the set of punctured handlebodies. We also show that any 3-manifold in T × I × I embeds in a punctured handlebody.

Uncountable collections of continua of dimension m embeddable in En are investigated, where the difference between m and n is not restricted to one. Collections of isometric copies of continua equivalent to Menger universal continua and collections of continua analogous to G. S. Young's Tn -sets are the main considerations.

It is shown in this paper that an immersion of a connected, closed n-manifold into another connected n-manifold is a diffeomorphism if and only if the induced homomorphism between the fundamental groups is surjective at some point. This is proved as a consequence of a more general assertion about topological spaces.