Smooth families and path components in the weak topology

Apply [L1] for formal data and [L2] for genuine data. If an open neighbourhood of $Q\times M$ rather than a product neighbourhood is given, compactness of $M$ and a finite product-neighbourhood cover of each $\{q\}\times M$ give a parameter neighbourhood of $q$ on which all the data are smooth; the union over $q\in Q$ gives the required open $W\supseteq Q$. For general smooth maps repeat the construction of [L2] with the immersion requirement removed: compactness gives a permitted value-error keeping interpolation in the tubular domain, parameter convolution gives the smooth approximant, and the cutoff fixes a neighbourhood of $Q$. No reference family, logarithm or complete metric is needed.

By the endpoint-preserving conclusion of [L2], every continuous path in $\mathrm{Imm}(M,N)$ joins regularly homotopic immersions. Conversely a regular homotopy is a continuous path by [F1]. This proves the component assertion with the prescribed endpoints unchanged.

For a based cubical family, precompose each coordinate with a smooth self-map of $[0,1]$ equal to zero near zero and one near one. This precomposition is homotopic to the identity by straight interpolation, preserves the boundary, and makes the family equal to its basepoint on a neighbourhood of the boundary. For a based spherical family, use a smooth self-map of the sphere homotopic to the identity relative to the basepoint and constant on a small neighbourhood of it: in a coordinate ball about that point replace the radial coordinate $r$ by a smooth function which is zero near zero and equals $r$ near the edge; extend by the identity outside the ball. Radial interpolation supplies the stated homotopy. These precompositions give smooth adjoint data near the relative set because the basepoint datum is a fixed smooth map or fixed smooth formal pair. Step 1.1 then supplies based smooth representatives.

To compare homotopies between smooth representatives, reparametrize the homotopy interval to be constant near its ends, and precompose the sphere or cube coordinates as in step 2.2. The resulting adjoint is smooth near the union of the end faces and the basepoint or boundary cylinder. Smooth it relative to that union using step 1.1. Its endpoints are the precomposed smooth representatives; each original smooth representative is joined to its precomposition by the smooth radial or cubical interpolation. Concatenating with smooth reparametrizations constant near the joining times gives a smooth based homotopy between the original representatives. Thus smooth representatives have exactly the same based homotopy classes as continuous ones.