On the commutator of unit quaternions and the numbers 12 and 24

The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this paper is to construct a concrete null-homotopy of the 12-th power of the commutator. Subsequently, we construct free S^3-actions on S^7 x S^3 whose quotients are exotic 7-sphere and give a geometric explanation for the order of the stable homotopy groups \pi_{n+3} (S^n). Intermediate results of perhaps independent interest are a construction of the octonions emphasizing the inclusion SU(3) \subset G_2, a detailed study of Duran's geodesic boundary map construction, and explicit formulas for the characteristic maps of the bundles G_2 \to S^6 and Spin(7) \to S^7.Comment: 27 pages, 2 references added, minor change

Chow points of C-orbits

We consider free algebraic actions of the additive group of complex numbers on a complex vector space X embedded in the complex projective space. We find an explicit formula for the map p that assigns to a generic point x in X the Chow point of the closure of the orbit through x. The properties Hausdorff quotient topology and proper action are equivalently characterized by the closure of the image of p in the closed Chow variety

Geometrically L^p-optimal lines of vertices of an equilateral triangle

We consider the distances between a line and a set of points in the plane defined by the L^p-norms of the vector consisting of the euclidian distance between the single points and the line. We determine lines with minimal geometric L^p-distance to the vertices of an equilateral triangle for all 1<= p<=\infty. The investigation of the L^p-distances for p\ne 1,2,\infty establishes the passage between the well-known sets of optimal lines for p=1,2,\infty. The set of optimal lines consists of three lines each parallel to one of the triangle sides for 1<= p < 4/3 and 2<p<=\infty and of the three perpendicular bisectors of the sides for 4/3<p<2. For p=2 and p=4/3 there exist one-dimensional families of optimal lines

Biquotient actions on unipotent Lie groups

We consider pairs (V,H) of subgroups of a connected unipotent complex Lie group G for which the induced VxH-action on G by multiplication from the left and from the right is free. We prove that this action is proper if the Lie algebra g of G is 3-step nilpotent. If g is 2-step nilpotent then there is a global slice of the action that is isomorphic to C^n. Furthermore, a global slice isomorphic to C^n exists if dim V = 1 = dim H or dim V = 1 and g is 3-step nilpotent. We give an explicit example of a 3-step nilpotent Lie group and a pair of 2-dimensional subgroups such that the induced action is proper but the corresponding geometric quotient is not affine

Fitting lines to points in the plane

We seek for lines of minimal distance to finitely many points in the plane. The distance between a line and a set of points is defined by the L^p-norm, 1\leq p\leq \infty, of the vector of vertical or orthogonal distances from the single points to the line. The known properties of optimal lines are deduced by elementary considerations and represented using a uniform language for the different choices to define the distance from a line to a set of points

Antiholomorphic involutions of spherical complex spaces

Let X be a holomorphically separable irreducible reduced complex space, K a connected compact Lie group acting on X by holomorphic transformations, theta : K -> K a Weyl involution, and mu : X -> X an antiholomorphic involution map satisfying mu(kx) = theta(k) mu(x) for x in X and k in K. We show that if the holomorphic functions on X form a multiplicity free K-module then mu maps every K-orbit onto itself. For a spherical affine homogeneous space X=G/H of the reductive group G (the complexification of K) we construct an antiholomorphic map mu with these properties

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

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. As applications we in particular determine the full isometry group of Sigma^7, classify all closed subgroups that act freely, determine the homotopy type of the corresponding orbit spaces, identify the Hirsch-Milnor involution in dimension 5 with the Calabi involution of W^5_3, and obtain explicit formulas for diffeomorphisms between the Brieskorn spheres W^5_3 and W^13_3 with standard Euclidean spheres

Presentations of the first homotopy groups of the unitary groups

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n >= 2 we supply n homotopic maps that each represent the (n-1)!-th power of a suitable generator of pi_2n(U(n)) = Z_{n!}. The product of these n commuting maps is the constant map to the identity matrix

An infinite family of Gromoll-Meyer spheres

We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.Comment: 13 page

Wiedersehen metrics and exotic involutions of Euclidean spheres

We provide explicit, simple, geometric formulas for free involutions rho of Euclidean spheres that are not conjugate to the antipodal involution. Therefore the quotient S^n/rho is a manifold that is homotopically equivalent but not diffeomorphic to RP^n. We use these formulas for constructing explicit non-trivial elements in pi_1 Diff(S^5) and pi_1 Diff(S^13) and to provide explicit formulas for non-cancellation phenomena in group actions.Comment: 17 pages, 5 figures, a QuickTime movie visualizing an exotic involution of S^5 is currently available at http://www.ruhr-uni-bochum.de/mathematik8/puttmann, revised version: corrections of typos, minor changes in the presentation, 1 reference adde