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.
