Is every homeomorphism from a compact manifold to itself isotopic to a homeomorphism with finitely many fixed points?
2
-
1$\begingroup$ Just a thought - maybe tricky to formalize - maybe you can take your homeomorphism, take a 'generic' vector field (i.e. one with finitely many critical points, which always exists) and flow along that a small amount to produce a new map whose only possible remaining fixed points are the original critical points. $\endgroup$– JoeCommented Sep 3, 2021 at 3:43
-
$\begingroup$ A variant of the question (maybe several here know the answer, if negative): is the set of self-homeomorphisms with finitely many fixed points dense in the self-homeomorphism group? If not, is its closure known? $\endgroup$– YCorCommented Oct 18, 2021 at 11:49
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
This is always true for surfaces. For dimension 3 and higher it is known that the homeomorphism may always be homotoped to a homeomorphism with finitely many fixed points. Since this question has been unanswered for a while I thought these references might be of interest.
For surfaces it follows from the main theorem of "Fixed points of surface diffeomorphisms" Boju Jiang and Jianhan Guo, Pacific J. Math. Vol. 160, No. 1, 1993. ProjectEuclid link
Quoting from the paper:
Theorem: (Jiang-Guo) Let $f: M \rightarrow M$ be a homeomorphism of a compact surface. When $M$ is closed, then $f$ is isotopic to a diffeomorphism with $N(f)$ fixed points, where $N(f)$ is its Nielsen number. When $M$ has boundary, $N(f)$ should be replaced by the relative Nielsen number $N(f;M,dM)$ defined by Schirmer.
Note that they prove something stronger that the number of fixed points can always taken to be the Nielsen number, for finitely fixed points it may have been known earlier (I am not an expert in this field so I don't know).
In dimension $\geq 3$ the statement up to homotopy follows from a theorem of Wecken (this is mentioned in the introduction of Jiang and Guo's article):
W] F. Wecken, Fixpunktklassen, I, Math. Ann., 117 (1941), 659-671; II, Math. Ann., 118 (1942), 216-234; III, Math. Ann., 118 (1942), 544-577:
Theorem (Wecken): Any self-homeomorphism $F$ of a manifold of dimension at least 3 may be homotoped to a map with exactly $N(F)$ fixed points.
I don't know whether it is known up to isotopy, since Jiang and Guo don't mention such a result I think it is reasonable to guess that it was open as of 1993 (although again I am not an expert so I cannot say for sure).
-
1$\begingroup$ Is this question sensitive to the difference between a self-homeomorphism and a self-diffeomorphism? The first theorem statement you give uses both in a way I find confusing -- but maybe that's the point? $\endgroup$ Commented Oct 18, 2021 at 11:54
-
1$\begingroup$ Yes, I think it is deliberate. Their result provides a "smoothing" in a given isotopy class to one with finitely many fixed point. I guess that the isotopy is in the topological category., $\endgroup$– Nick LCommented Oct 18, 2021 at 12:03