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Connected sums of codimension two locally flat submanifolds
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 16 January 2023, pp. 1-8
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- Article
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Let $X$
and $Y$
be oriented topological manifolds of dimension $n\!+\!2$
, and let $K\! \subset \! X$
and $J \! \subset \! Y$
be connected, locally-flat, oriented, $n$
–dimensional submanifolds. We show that up to orientation preserving homeomorphism there is a well-defined connected sum $(X,K)\! \mathbin {\#}\! (Y,J)$
. For $n = 1$
, the proof is classical, relying on results of Rado and Moise. For dimensions $n=3$
and $n \ge 6$
, results of Edwards-Kirby, Kirby, and Kirby-Siebenmann concerning higher dimensional topological manifolds are required. For $n = 2, 4,$
and $5$
, Freedman and Quinn's work on topological four-manifolds is required along with the higher dimensional theory.
