arXiv subjects
Publications and source records attributed to Thomas Puettmann.
We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups SO(4L+2), L >= 1, with degree -3, of SO(8), SO(14) and SO(26) with degree -5 each, of SO(10) with degree -7, and of SO(14) with degree -11 by exhibiting linear solutions to non-linear singular boundary value problems.
We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group $G$ with principal isotropy group $H$ and cohomogeneity $k$ such that $k - (\rank G - \rank H)\le 5$. Moreover, we prove that the Euler characteristic of a compact Riemannian manifold $M^{4l+4}$ or $M^{4l+2}$ with positive sectional curvature is positive if $M$ admits an effective isometric action of a torus $T^l$, i.e., if the symmetry rank of $M$ is $\ge l$.
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 π_{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.
We construct natural selfmaps of compact cohomgeneity one manifolds with finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields in certain cases relations between the order of the Weyl group and the Euler characteristic of a principal orbit. We apply our construction to the compact Lie group SU(3) where we extend identity and transposition to an infinite family of selfmaps of every odd degree. The compositions of these selfmaps with the power maps realize all possible degrees of selfmaps of SU(3).
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.
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.
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.