Measure of the image of functions homotopy equivalent to the identity on a Riemann surface

Question

Let $S_g$ be a closed orientable surface of genus $g\ge 2$. Let $f:S_g\rightarrow S_g$ be a continuous function such that $f_*:\pi_1(S_g)\rightarrow \pi_1(S_g)$ is equal to the identity on $\pi_1(S_g)$ (or equivalently $f$ is homotopy equivalent to $Id:S_g\rightarrow S_g$).

• Is it true that $S_g\setminus f(S_g)$ is a set of measure $0$?

Call $V\subset S_g$ a maximal subset such that $f:V\rightarrow S_g$ is injective.

• Is it true that $S_g\setminus V$ and $S_g\setminus f(V)$ are sets of measure $0$ for every $V$ maximal? Does there exist a maximal $V$ such that $S_g\setminus V$ or $S_g\setminus f(V)$ are sets of measure zero?

Answer 1

1. The map $f$ has degree 1 and, hence, is surjective. (The best reference I know is Guillemin and Pollack "Differential Topology". Given two closed oriented connected manifolds $M, N$ of the same dimension and a smooth map $f: M\to N$, $deg(f)$ can be defined as the sum $$ \sup_{x\in f^{-1}(y)} (sign(J_x(f)) $$ where $J_x(f)$ is the Jacobian determinant of $f$ at $x$ and $y\in N$ is a regular value of $f$, i.e. a point such that $J_x(f)\ne 0$ for all $x\in f^{-1}(y)$. If $f$ is not surjective, then any $y\notin f(M)$ is a regular value of $f$ and, hence, $deg(f)=0$. However, degree is a homotopy-invariant. If $f: M\to M$ is homotopic to the identity, then $deg(f)=1$. Thus, every self-map homotopic to the identity is surjective.)

2. There is no such a thing as the subset $V$ where $f$ is injective, there are many subsets $V$ such that the restrictions of $f$ to $V$ are injective. Maybe you want a maximal subset $V$ with this property.

Edit. If we make no assumptions about topology of $V$ then every such $V$ will have the property that $f: V\to S_g$ is a bijection. Here is an example showing that $V$ need not have full measure in $S_g$. Let $D\subset S_g$ be a closed disk; suppose that $f|_D$ is constant, equal to $q\in S_g$ and $f|_{S_g - D}$ is a homeomorphism to $S_g -\{q\}$. Then any maximal $V$ will equal $(S_g - D) \cup\{p\}$ where $p$ is a point in $D$.

One more thing: One can construct (using Peano-type curves) maps $f: S_g\to S_g$ homotopic to the identity and maximal injective subsets $V\subset S_g$, such that $V$ is a subset of a smooth arc, in particular, noweher dense and of measure zero.