# dg.differential geometry – Homotoping diffeomorphism to a \$J\$-holomorphic one

Let $$M$$ be a closed simply-connected smooth manifold. Assume $$M$$ admits at least one almost complex structure.

Is any diffeomorphism $$Mto M$$ homotopic as a continuous map to a $$J$$-holomorphic diffeomorphism $$Mto M$$ where $$J$$ is some almost complex structure?