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Some topologically locally-flat surfaces in the complex projective plane. (English) Zbl 0575.57003

The author gives examples of the following: Theorem 1. For every integer \(n\geq 6\), there exists in the homology class \(n[{\mathbb{C}}{\mathbb{P}}^ 1]\in H_ 2({\mathbb{C}}{\mathbb{P}}^ 2;{\mathbb{Z}})\) a topologically locally flat embedded surface of genus strictly less than that of a non-singular complex algebraic curve of degree n (i.e. (1/2)(n-1)(n-2)).
Theorem 2. For every pair (m,n) of integers greater than or equal to 5 (except possibly (5,5)), there is a topologically locally flatly embedded surface in the 4-disc with boundary a torus link \(O\{\) m,n\(\}\) of type (m,n) and genus strictly less than the (classical) genus of \(O\{\) m,n\(\}\).
Note that topologically locally flat surfaces cannot be replaced in general by differentiable or P.L. locally flat surfaces. The construction used in theorems 1 and 2 relies on a result of Freedman namely: Let \(K\subset \partial D^ 4\) be a smooth knot with trivial Alexander polynomial. Then K bounds a topologically locally flat disc in \(D^ 4\).
Reviewer: R.A.Fenn

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R95 Realizing cycles by submanifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N45 Flatness and tameness of topological manifolds
57N35 Embeddings and immersions in topological manifolds
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