Every homeomorphism isotopic to one with finitely many fixed points Is every homeomorphism from a compact manifold to itself isotopic to a homeomorphism with finitely many fixed points?
 A: 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).
