Universität Augsburg
Institut für Mathematik

Like spheres many other compact symmetric spaces have "poles", "meridians" (shortest geodesics between poles) and "equators", correctly "centrioles" (midpoint set components for meridians). Sometimes the maps from the $k$-sphere into such space sending poles to poles can be deformed onto maps sending meridians to meridians. These deformed maps are geodesic suspensions over maps from the equator $S^{k-1}$ to the centriole. This is the main construction step in Milnor's "Morse Theory". It implies Bott's periodicity theorem for the orthogonal group since the 8th iterated centriole of $SO_n$ is $SO_{n/16}$. We pushed forward Milnor's approach in two ways:

1. We exhaust $S^k$ by going all the way down to $S^0$. Thus we deform each map $S^k\to SO_n$ into a normal form given by a representation of the Clifford algebra $Cl(R^k)$ on $R^n$, taken from the $k$-fold geodesic suspension. In turn, Maps $S^k\to SO_n$ are clutching maps of vector bundles over $S^{k+1}$.

2. Milnor's methods allow dependence on additional parameters, even locally. Thus we may replace the sphere $S^k$ by certain sphere bundles over finite CW-complexes. After passing to topological K-theory these results imply classical Bott–Thom isomorphism theorems.