Quinn theorem

In differential topology in mathematics, the Quinn theorem is a result about topological 4-manifolds. It shows the importance of the topological property of compactness for smoothability, hence if there exists a compatible smooth structure. The Quinn theorem is named after Frank Quinn, who proved it in 1984.

Formulation

Every non-compact topological 4-manifold is smoothable.[1] A counterexample is the E8 manifold, which is a compact and non-smoothable topological 4-manifold. A more general formulation is: A topological 4-manifold has a smooth structure in the complement of any closed set with at least one point in each compact component.[2][3]

See also

Literature

  • Quinn, Frank (1984). "Smooth structures on 4-manifolds". American Mathematical Society. Contemp. Math. 35: 473–479.
  • Freedman, Michael; Quinn, Frank (1990). "8.2 Smoothing open 4-manifolds". Topology of 4-Manifolds. p. 116. ISBN 9780691632346.
  • Scorpan, Alexandru (2005). The Wild World of 4-Manifolds. Mathematical Sciences Research Institute Publications. Vol. 1. American Mathematical Society. ISBN 978-1-4704-6861-3.

References

  1. ↑ Freed, Daniel; Uhlenbeck, Karen (1984). Instantons and Four-Manifolds. Mathematical Sciences Research Institute Publications. Vol. 1. Springer. Theorem 1.4 on p. 25. doi:10.1007/978-1-4613-9703-8. ISBN 978-1-4613-9705-2.
  2. ↑ Freedman & Quinn 1990, p. 116
  3. ↑ Scorpan 05, p. 219
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