The Topological Fundamental Group and Hoop Earring Spaces · arXivDesk
0910.3685Oct 19, 20099 pages This paper has been withdrawn by the author due to much more general results appearing in the paper "The topological fundamental group and free topological groups" which includes the results in this paper. This paper, includes a sketch of the main proof (citing a thesis in progress) whereas a full proof is given in the mentioned paper.
The Topological Fundamental Group and Hoop Earring Spaces
The topological fundamental group π1top is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased space X, we compute the topological fundamental group of the "hoop earring" space of X, which is the reduced suspension of X
Nearby in the stack
with disjoint basepoint. We do so by factorizing the quotient map
Ω(ΣX+,x)→π1top(ΣX+,x)
through a free topological monoid with involution
M(X)
such that the map
M(X) π₁^top(ΣX₊,x)
is also a quotient map.
π1top(ΣX+,x)
is T1 and an embedding
X π₁^top(ΣX₊,x)
illustrates that
π1top(ΣX+,x)
is not a topological group when
X
is not regular. These hoop earring spaces provide a simple class of counterexamples to the claim that
π1top
is a functor to the category of topological groups.