Topological Classification of Multiaxial $U(n)$-Actions

Min Yan
Monday, July 22, 2013 - from 17:00 to 17:25
A manifold under the action of the unitary group $U(n)$ is multiaxial if all the isotropy groups are unitary subgroups. Such manifolds are often modeled on $k{\mathbb C}^n\oplus j{\mathbb R}$, where ${\mathbb C}^n$ has the canonical $U(n)$-action and ${\mathbb R}$ has trivial $U(n)$-action. One of the major achievements about multiaxial manifolds was M. Davis and W.C. Hsiang's concordance classification in late 1970s for the smooth category and under the assumption $k\le n$. We study the structure set $S_{U(n)}(M)$ of a multiaxial $U(n)$-manifold $M$, which is the homeomorphism classes of the topological $U(n)$-manifolds equivariantly homotopy equivalent to $M$. We show that $S_{U(n)}(M)$ can be decomposed into simpler structure sets. We discuss the implication of the decomposition and compute explicitly for the case $M$ is the canonical representation sphere. All the results do not assume $k\le n$.