Loading AI tools
On when elements of the 2nd homotopy group of a 3-manifold can be embedded spheres From Wikipedia, the free encyclopedia
In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.
One example is the following:
Let be an orientable 3-manifold such that is not the trivial group. Then there exists a non-zero element of having a representative that is an embedding .
The proof of this version of the theorem can be based on transversality methods, see Jean-Loïc Batude (1971).
Another more general version (also called the projective plane theorem, and due to David B. A. Epstein) is:
Let be any 3-manifold and a -invariant subgroup of . If is a general position map such that and is any neighborhood of the singular set , then there is a map satisfying
quoted in (Hempel 1976, p. 54).
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.