On totally real isotopy classes (Q2778028)

Let \(M^n\) be an \(n\)-dimensional manifold without boundary and \((W^{2n},J)\) a \(2n\)-dimensional almost complex manifold. Denote by \(I_{rn}\) the space of totally real immersions \(j\) of \(M^n\) into \(W^{2n}\) and by \(E\) the space of ordinary embeddings. Let these spaces be provided with compact-open topology. The present paper supplies a partial answer to the following questions: If two totally real embeddings \(j_0\) and \(j_1\) that belong to the same component of \(E\) and of \(I_{rn}\) are given, is it possible to find a totally real isotopy joining \(j_0\) and \(j_1\)?