I'm pretty sure this has an easy solution, but I can't seem to find it.
Let X be a contractible 2-dimensional CW-complex, let gamma be an embedded loop in X, and let f:D2rightarrowX be an embedding of a disc in X which maps the boundary of D to gamma.
My question is the following. Let f′:D2rightarrowX be a continuous map of a disc into X which takes the boundary of D to gamma. Must we then have f(D2)subsetf′(D2) ? I'm pretty sure that the answer is yes, but I can't seem to prove it.
Of course, this has an obvious generalization to higher dimensional complexes, and I'd be interested in that too.
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