  Consider a family $\mathcal F$ of $C_{2r+1}$-free graphs, where $r\geq 2$.
Suppose that each graph in $\mathcal F$ has minimum degree linear in its number
of vertices. Thomassen showed that such a family has bounded chromatic number,
or, equivalently, that all graphs in $\mathcal F$ are homomorphic to a complete
graph of bounded size. Considering instead homomorphic images which are
themselves $C_{2r+1}$-free, we construct a family of dense $C_{2r+1}$-free
graphs with no $C_{2r+1}$-free homomorphic image of bounded size. This provides
the first nontrivial lower bound on the homomorphism threshold of odd cycles of
length at least 5 and answers a question of Ebsen and Schacht.
  Our proof introduces a new technique to describe the topological structure of
a graph. We establish a graph-theoretic analogue of homotopy equivalence, which
allows us to analyze the relative placement of odd closed walks in a graph.
This notion has unexpected connections to the neighborhood complex, leading to
multiple interesting questions.
