Gromoll Meyer sphere

In mathematics , the Gromoll-Meyer sphere is an example of an exotic sphere; H. a differential structure on a sphere that is not equivalent to the standard differential structure . It creates the group of 7-dimensional homotopy spheres and was the first example of an exotic sphere with a metric of nonnegative sectional curvature .

history

An exotic sphere is a differentiable manifold that is homeomorphic but not diffeomorphic to the unit sphere in is. John Milnor was 1956, the first examples of exotic spheres, which he called - bundle over engineered. He proved that there are 28 different differential structures on the 7-dimensional sphere and that the group of 7-dimensional homotopy spheres modulo h-cobordism isomorphic to ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle S^{3}}$ ${\displaystyle S^{4}}$

${\displaystyle \mathbb {Z} /28\mathbb {Z} }$

is. The bundles above (so-called Milnor spheres ) correspond to the 21 elements ${\displaystyle S^{3}}$ ${\displaystyle S^{4}}$

${\displaystyle \left\{0,1,\ldots ,10,18,\ldots ,28\right\}}$,

where the bundle with Euler number and Pontryagin number is the element ${\displaystyle 1}$ ${\displaystyle 8k+2}$

${\displaystyle k\mod 28}$

in corresponds. Egbert Brieskorn showed in 1966, found that the Milnor spheres as tangles of singularities of hypersurfaces in can be described, namely as a section of the hypersurface ${\displaystyle \mathbb {Z} /28\mathbb {Z} }$ ${\displaystyle \mathbb {C} ^{5}}$

${\displaystyle a^{2}+b^{2}+c^{2}+d^{3}+e^{6k-1}=0}$

with a small sphere around the zero point. In 1974 Gromoll and Meyer gave a construction of the producer (i.e. the corresponding element of the group of homotopy spheres ) as the biquotient of the group ${\displaystyle k=1}$

${\displaystyle Sp(2)}$

and in particular found the first example of a Riemannian metric of nonnegative sectional curvature on an exotic sphere. Grove and Ziller proved in 2000 that the other Milnor spheres also have a metric of nonnegative sectional curvature. In 2010 Duran, Püttmann and Rigas gave a construction of all exotic 7 spheres derived from the Gromoll-Meyer construction.

construction

Let it be the compact symplectic group , i.e. H. the group of the canonical inner product on the 2-dimensional vector space quaternionic maintaining -linear pictures, and let the group of quaternions of standard . Then acts on by ${\displaystyle Sp(2)}$ ${\displaystyle \mathbb {H} ^{2}}$ ${\displaystyle \mathbb {H} }$ ${\displaystyle Sp(1)\simeq S^{3}}$ ${\displaystyle 1}$ ${\displaystyle Sp(1)\times Sp(1)}$ ${\displaystyle Sp(2)}$

${\displaystyle (q_{1},q_{2})\circ Q:={\begin{pmatrix}q_{1}&0\\0&q_{1}\end{pmatrix}}Q{\begin{pmatrix}{\overline {q}}_{2}&0\\0&1\end{pmatrix}}}$.

This effect is free with a quotient . The diagonal is particularly effective ${\displaystyle S^{4}}$

${\displaystyle \Delta =\left\{(q,q)\in Sp(1)\times Sp(1)\right\}}$

frei auf and Gromoll and Meyer proved that the quotient is diffeomorphic to the Milnor sphere with . ${\displaystyle Sp(2)}$ ${\displaystyle \Delta \backslash Sp(2)}$ ${\displaystyle k=1}$

From O'Neill's formula it follows that has nonnegative section curvature and positive Ricci curvature . The metric can be deformed so that the cutting curvature becomes positive almost everywhere. ${\displaystyle \Delta \backslash Sp(2)}$

literature

• John Milnor: On manifolds homeomorphic to the 7-sphere. In: Ann. of Math. 2. Volume 64, 1956, pp. 399-405. (pdf)
• Detlef Gromoll, Wolfgang Meyer: An exotic sphere with nonnegative sectional curvature. In: Ann. of Math. 2nd Volume 100, 1974, pp. 401-406. (pdf)
• Karsten Grove, Wolfgang Ziller: Curvature and symmetry of Milnor spheres. In: Ann. of Math. 2nd Volume 152, no. 1, 2000, pp. 331-367. (pdf)
• Carlos Durán, Thomas Püttmann, A. Rigas: An infinite family of Gromoll-Meyer spheres. In: Arch. Math. (Basel). Volume 95, no. 3, 2010, pp. 269-282. (pdf)

Web links

• M. Joachim, DJ Wraith: Exotic spheres and curvature. In: Bull. Amer. Math. Soc. (NS). Volume 45, no. 4, 2008, pp. 595-616. (pdf)
• Exotic Spheres (Manifold Atlas)