algebraic-topologygeneral-topology
When I read a text book, I encountered the sentence
"The modular group of genus $n$ is the group of isotopy classes of degree $1$ self-homeomorphism of a closed oriented surface of genus $n$".
Is "degree $1$" equivalent to "orientation preserving homeomorphism"?
Your conclusion isn't true. Here is an example of a map homotopic to the identity:
Pick a disk around the punctured point and declare the map outside this disk to be the identity. Pick a chart so this is the unit disk. Then define the map inside the disk to rotate the points by $1-r$ where r is the distance to the origin.
A homotopy to the identity is given by $H(x,t)$ is the identity if $x$ is outside the disk and $H(x,t)$ rotates the points by $(1-t)(1-r)$.
I think the issue with your reasoning is your statement that it is $\mathbb{R}$-linear and stabilize the lattice. I can't see any reason either of these would be true.
A curve is a continuous map $[0,1]\to M$, where $M$ is the manifold on which you're considering it. In the same way one can define a point (a constant curve) and a boundary component if the manifold is of dimension 2 (and therefore the boundary is of dimension 1, meaning that it is a disjoint union of curves). An example of essential closed curve is given in the following picture taken from Basic results on braid groups by Juan González-Meneses.
That depends first of all on how you define orientations. Since I don't want to assume any smooth structure, I'll say it in terms of local homology. An orientation on a manifold $M$ of dimension $n$ is a choice of generator $[M]_x$ of $H_n(M,M-x;\mathbb{Z})$ for each $x\in M$ in such a way that this choice is continuous (See Definition 3.1, and the others if you want another notion). A homeomorphism is orientation preserving if the induced maps on local homology takes orientation to orientations.
Best Answer
Well, no. Maps of degree 1 need not be self-homeomorphisms, since they can easily fail to be injective. Imagine taking a circle and looping a little piece of it over itself. This is isotopic to the identity, but is certainly not a self-homeomorphism.