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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.

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  • $\begingroup$ What do you mean by low dimensional version of invarience of domain? Can you please state is explicitly? $\endgroup$ Dec 11 '16 at 16:10
  • $\begingroup$ 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. $\endgroup$ Dec 11 '16 at 16:33
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This enlarges on my comment.

Munkres' book "Topology: a first course" (1975) gave a proof of the Jordan Curve Theorem using the fundamental group. One part of the proof involved a covering space argument to get information on the fundamental group of a union of spaces without assumptions of path connectedness of intersections.

I adapted this proof but using the fundamental groupoid on a set of base points, and this proof was published in the 1988 edition of the (1968) book which is now available in its 2006 edition as Topology and Groupoids. This proof related parts of the argument to the Phragmen-Brouwer Property, which is of independent interest.

Omar Antolin-Camarena noticed a small gap in that proof and we published a correction in the paper cited in the comment, which also has wider references than in my book.

No relation of these arguments to Invariance of Domain is given.

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