Approximating topological surfaces in 4-manifolds

Venema, Gerard A.

doi:10.1090/S0002-9947-1981-0607105-3

Approximating topological surfaces in $4$-manifolds
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by Gerard A. Venema

Trans. Amer. Math. Soc. 265 (1981), 35-45

DOI: https://doi.org/10.1090/S0002-9947-1981-0607105-3

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Abstract:

Let ${M^2}$ be a compact, connected $2$-manifold with $\partial {M^2} \ne \emptyset$ and let $h:{M^2} \to {W^4}$ be a topological embedding of ${M^2}$ into a $4$-manifold. The main theorem of this paper asserts that if ${W^4}$ is a piecewise linear $4$-manifold, then $h$ can be arbitrarily closely approximated by locally flat PL embeddings. It is also shown that if the $4$-dimensional annulus conjecture is correct and if $W$ is a topological $4$-manifold, then $h$ can be arbitrarily closely approximated by locally flat embeddings. These results generalize the author’s previous theorems about approximating disks in $4$-space.

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