# Jordan curve theorem and invariance of domain via fundamental group.

It is well-known that one of the most successful ways to attack theorems like Brouwer fixed point theorem, Jordan curve theorem etc is via algebraic topology, more explicitly computing some homology groups of the objects involved in the situations.

When in low dimensions, there are some proofs which can be reduced to handling the fundamental group instead. For instance, Brouwer fixed point theorem on $D^2$ can be proved in a very similar way to how it is usually done via homology (using functoriality and the non-triviality of a group associated to $S^n$, $n=1$ in the case of $D^2$). There is also an argument which proves Borsuk-Ulam theorem for maps $S^2 \to S^2$.

My question is: are there methods to prove low-dimensional versions (of course, not "trivial" ones, like the invariance of domain for dimension $1$) of invariance of domain and Jordan curve theorem that use the fundamental group alone? Thanks in advance.

• What do you mean by low dimensional version of invarience of domain? Can you please state is explicitly? Dec 11 '16 at 16:10
• This paper groupoids.org.uk/pdffiles/brouwer-cor-fin.pdf proves the Jordan Curve Theorem using methods of the fundamental groupoid on a set of base points. It links this result with work on the Phragmen-Brouwer Property. Dec 11 '16 at 16:33