Whitehead manifold

In mathematics , the Whitehead manifold is an example of a contractible 3-manifold that is not homeomorphic to Euclidean space . ${\displaystyle \mathbb {R} ^{3}}$

JHC Whitehead published a proof of the Poincaré conjecture in 1934 , in which he first wanted to have proven that every contractable 3-manifold is homeomorphic to , from which he obtained the Poincaré conjecture (every simply connected closed 3-manifold is homeomorphic to ). The following year he discovered a flaw in his proof and the example of the Whitehead manifold. This is contractible, but not simply connected in infinity , which means that it cannot be homeomorphic and refutes his first claim. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle S^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$

construction

Construct a sequence of full gates embedded in the 3-sphere as follows. ${\displaystyle W_{i}}$ ${\displaystyle S^{3}}$

1st step: is an untied full torus in . ${\displaystyle W_{1}}$ ${\displaystyle S^{3}}$

2nd step: is embedded in so that the core of forms a Whitehead loop with the meridian of . ${\displaystyle W_{2}}$ ${\displaystyle W_{1}}$ ${\displaystyle W_{2}}$ ${\displaystyle W_{1}}$

...

i. Step: is embedded in so that the core of forms a Whitehead link with the meridian of . ${\displaystyle W_{i+1}}$ ${\displaystyle W_{i}}$ ${\displaystyle W_{i+1}}$ ${\displaystyle W_{i}}$

...

The Whitehead manifold is the complement of the intersection in , or equivalently the union with . ${\displaystyle \bigcap _{i\in \mathbb {N} }W_{i}}$ ${\displaystyle S^{3}}$ ${\displaystyle \bigcup _{i\in \mathbb {N} }N_{i}}$ ${\displaystyle N_{i}=S^{3}\setminus W_{i}}$

Topological properties

The Whitehead manifold is contractible and , ${\displaystyle W}$ ${\displaystyle W\times \mathbb {R} \cong \mathbb {R} ^{4}}$

It is not simply coherent in infinity. Your one-point compactification is the quotient of when all points are identified with each other. ${\displaystyle S^{3}}$ ${\displaystyle \bigcap _{i\in \mathbb {N} }W_{i}}$

It is the union of two copies of the whose intersection is also homeomorphic to . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$

Differential geometry

The Whitehead manifold does not have a complete Riemannian metric of positive scalar curvature .

Individual evidence

1. ^ JHC Whitehead: Certain theorems about three-dimensional manifolds (I) , Quarterly Journal of Mathematics 5, 308-320 (1934)
2. ↑ JHC Whitehead: A certain open manifold whose group is unity , Quarterly Journal of Mathematics 6, 268-279 (1935)
3. ↑ David Gabai : The Whitehead manifold is a union of two Euclidean spaces , Journal of Topology 4, 529-534 (2011)
4. ↑ J. Wang: Contractible 3-manifold and Positive scalar curvature , ArXiv