Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $M$ be a closed smooth manifold. A generic diffeomorphism $\phi: M\rightarrow M$ has non-degenerate fixed points, i.e. the intersections of its graph with the diagonal in $M\times M$ are all transverse. In a generic one-parameter family $(\phi_t)$ of diffeomorphisms (say starting at the identity) we cannot request that the fixed points of $\phi_t$ be non-degenerate for all $t$. Nonetheless we can arrange that degenerate fixed points occur only for finitely many values of $t$ and that they be of "birth-death" type. In particular, the fixed point set of each $\phi_t$ is still isolated.

How "non-generic" is it for a diffeomorphism to have a fixed point set that "carries topology", i.e. not only its fixed point set $F$ is infinite but it has non-trivial homology and the map $H_\ast(F)\rightarrow H_\ast(M)$ is non-trivial at some positive degree? The reasoning above shows that the codimension of these diffeomorphisms in $Diff(M)$ is at least 2. Is it finite/very high/infinte?

As a particular example, let $M=\mathbb{CP}^n$ and consider the set $\mathcal {E}$ of diffeomorphisms whose fixed point set contains a line, i.e. the restriction of a generator $u\in H^2(\mathbb{CP}^n;\mathbb Q)$ to its fixed point set is non-zero. Does $\mathcal{E}$ have codimension greater than $2$ in $Diff(M)$? Analogous questions can be asked in the volume-preserving or symplectic cases.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.