Is the fixed point set of an involution on a topological manifold a submanifold? Let $f:X\to X$ be a homeomorphism of a topological manifold with $f^2=\mathrm{id}$.
Is each connected component of $\{x\in X \mid f(x)=x\}$ a topological submanifold?
 A: No, the connected components of the fixed points of $f$ need not be a manifold. I can give an example of a homeomorphism $f\colon \mathbb{R}^4\to\mathbb{R}^4$ with $f^2=\rm id$, whose set of fixed points is connected (in fact, contractible), but is not a manifold.
Let $X=B\times\mathbb{R}$, where $B$ is Bing's dogbone space, and define $f\colon X\to X$ by $f(b,r)=(b,-r)$. This is clearly a homeomorphism with $f^2=\rm id$ and $\lbrace x\in X\colon f(x)=x\rbrace=B\times\lbrace0\rbrace\cong B$. It is known that $X$ is homeomorphic to $\mathbb{R}^4$, but the dogbone space $B$ is not a topological manifold. Also, $B$ is contractible, as its product with $\mathbb{R}$ is contractible.
Note: Bing's construction of the dogbone space was published in May 19571, and the fact that its product with $\mathbb{R}$ is $\mathbb{R}^4$ was demonstrated in 19592. Eilenberg's list of open problems linked to by Willie Wong in the comment to the question dates back to April 19493. So, it is entirely plausible that the problem was indeed open when Eilenberg published his list.

1 A Decomposition of $E^3$  into Points and Tame Arcs Such That the Decomposition Space is Topologically Different from $E^3$, R. H. Bing, Annals of Mathematics Second Series, Vol. 65, No. 3 (May, 1957), pp. 484-500. (Link)
2 The Cartesian Product of a Certain Nonmanifold and a Line is $E^4$, R. H. Bing, Annals of Mathematics Second Series, Vol. 70, No. 3 (Nov., 1959), pp. 399-412. (Link)
3 On the Problems of Topology, Samuel Eilenberg, Annals of Mathematics Second Series, Vol. 50, No. 2 (Apr., 1949), pp. 247-260. (Link)
