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In this brief note, we investigate the ${\mathbb{C}\mathbb{P}}^2$-genus of knots, i.e., the least genus of a smooth, compact, orientable surface in ${\mathbb{C}\mathbb{P}}^2\setminus <\#comment/>\stackrel{˚<\#comment/>}{B^4}$bounded by a knot in $S^3$. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the ${\mathbb{C}\mathbb{P}}^2$-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in ${\mathbb{C}\mathbb{P}}^2\mathrm{\#<\#comment/>}{\mathbb{C}\mathbb{P}}^2$. 

Abstract A subset E of a metric space X is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $$\mathbb{R}^n$$ for some n , sending E to a starlike set. A subset $$E\subset X$$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $$\{E_i\}_{i=0}^{N+1}$$ such that $$E_{i}/E_{i+1}\subset X/E_{i+1}$$ is starlike-equivalent for each i and $$E_{N+1}$$ is a point. A decomposition $$\mathcal{D}$$ of a metric space X is said to be recursively starlike-equivalent, if there exists $$N\geq 0$$ such that each element of $$\mathcal{D}$$ is recursively starlike-equivalent of filtration length N . We prove that any null, recursively starlike-equivalent decomposition $$\mathcal{D}$$ of a compact metric space X shrinks, that is, the quotient map $$X\to X/\mathcal{D}$$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4-manifolds, including the four-dimensional Poincaré conjecture. 

Abstract 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.