# Wiedersehen metrics and exotic involutions of Euclidean spheres

@inproceedings{Abresch2005WiedersehenMA, title={Wiedersehen metrics and exotic involutions of Euclidean spheres}, author={Uwe Abresch and Carlos E. Dur{\'a}n and Thomas Puettmann and Alcib{\'i}ades Rigas}, year={2005} }

Abstract We provide explicit, simple, geometric formulas for free involutions ρ of Euclidean spheres that are not conjugate to the antipodal involution. Therefore the quotient Sn/ρ is a manifold that is homotopically equivalent but not diffeomorphic to . We use these formulas for constructing explicit non-trivial elements in π1 Diff(S 5) and π1 Diff(S 13) and to provide explicit formulas for non-cancellation phenomena in group actions.

#### 16 Citations

On the Geometry of Some Equivariantly Related Manifolds

- Mathematics
- International Mathematics Research Notices
- 2018

We provide geometric realizations of both classical and “exotic” $G$-manifolds such as spheres, Kervaire manifolds, bundles over spheres, homogeneous spaces, and connected sums among them. As an… Expand

Pulling back the Gromoll-Meyer construction and models of exotic spheres

- Mathematics
- 2010

We generalize the construction of the Milnor sphere M 2,−1 in [10] through a pull-back procedure and apply it to exhibit explicit nontrivial elements on some equivariant homotopy groups and… Expand

Isoparametric functions on exotic spheres

- Mathematics
- 2015

Abstract This paper extends widely the work in [11] . Existence and non-existence results of isoparametric functions on exotic spheres and Eells–Kuiper projective planes are established. In… Expand

An exotic sphere with positive curvature

- Mathematics
- 2014

A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a… Expand

A MINIMAL BRIESKORN 5-SPHERE IN THE GROMOLL-MEYER SPHERE AND ITS APPLICATIONS

- Mathematics
- 2006

We recognize the Gromoll-Meyer sphere � 7 as the geodesic join of a simple closed geodesic and a minimal subsphere � 5 � � 7 , which can be equivariantly identified with the Brieskorn sphere W 5 3 .… Expand

Suspending the Cartan embedding of HP n through spindles and generators of homo- topy groups

- 2010

We provide an equivariant suspension of the Cartan embedding of the symmetric space S → HP ↪→ Sp(n+ 1); this construction furnishes geometric generators of the homotopy group of π4n+6Sp(n+ 1). We… Expand

Equivariant homotopy and deformations of diffeomorphisms

- Mathematics
- 2007

Abstract We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres… Expand

A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications

- Mathematics
- 2006

We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a simple closed geodesic and a minimal subsphere Sigma^5, which can be equivariantly identified with the Brieskorn sphere W^5_3.… Expand

Pulling back the Gromoll-Meyer construction and models of exotic spheres

- Mathematics
- 2010

Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free… Expand

ALMOST NONNEGATIVE CURVATURE ON SOME FAKE 6- AND 14-DIMENSIONAL PROJECTIVE SPACES

- Mathematics
- Bulletin of the Australian Mathematical Society
- 2016

We apply the lifting theorem of Searle and the second author to put metrics of almost nonnegative curvature on the fake $\mathbb{R}P^{6}$ s of Hirsch and Milnor and on the analogous fake… Expand

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