Metastable approximation of maps by embeddings

In the simultaneous case replace $C$ by $C\cup \mathrm{\partial }N$. In the unrestricted case take $C=\mathrm{\varnothing }$. If $N$ is empty the claim is vacuous. Under the relative embedding hypothesis compactness supplies nested closed neighbourhoods $D_0\subset \mathrm{int}{D}_1$ of $C$ on which $f$ is an embedding. To see this, injectivity of the differential gives local embedding neighbourhoods about every point of $C$. If no smaller neighbourhood were globally injective, distinct pairs in shrinking neighbourhoods would converge by compactness to a pair in $C$ with equal images. Injectivity on $C$ forces the limits equal; one local embedding neighbourhood then excludes the pairs. In the independent boundary-only case use a source collar from [F9] and choose a smooth normal field to the embedded boundary inside $M$: after embedding $M$ in Euclidean space, project constant-vector parameters onto this normal bundle. The resulting section-evaluation is a submersion onto the fibre, and parametric transversality avoids its zero section because the normal rank $m-n+1$ exceeds $n-1$. The embedded boundary is compact, hence closed in $M$, so [F1] applies to it; a nonzero field in its normal bundle gives an embedded collar in that tube. Replace $f$ near its boundary by this collar extension, through a homotopy fixed on the boundary: on a small collar both maps are close to the same boundary value, and a target tubular retraction of their Euclidean linear interpolation gives the homotopy, with a cutoff on a slightly larger collar. Thus the relative argument applies with $C=\mathrm{\partial }N$ and a protected collar. If the protected neighbourhood covers $N$, the prepared map is already an embedding; otherwise proceed with the adjustable core.

Embed $M$ properly in Euclidean space and use a smooth tubular retraction $R$. Cover the compact core outside $\mathrm{int}{D}_1$ by finitely many source charts with bumps supported off $D_0$ and equal to one on smaller charts. For each bump independently multiply the constant function and all $n$ coordinate functions by every ambient coordinate vector, with independent parameters. At any point of a smaller chart constants span value variations; subtracting their appropriate multiples from the coordinate profiles gives functions vanishing at that point whose derivatives span the $n$ independent derivative columns. Composing the Euclidean perturbations with $R$ gives independent value and derivative variations, since $dR$ maps onto the target tangent space. Hence the local 1-jet evaluation is a submersion on the adjustable core at parameter zero, and remains so in a sufficiently small parameter ball by compactness. It is not asserted to be a submersion on the protected collar, where immersion already holds.

For a rank-$r$ matrix of size $m$ by $n$, a chart with an invertible $r$-minor writes the rank-$r$ stratum as the vanishing of the Schur-complement block. Its codimension is $(m-r)(n-r)$, least $m-n+1$ for $r<n$. Apply parametric transversality to these finitely many rank strata in local jet charts, restricting to their open matrix-chart domains; a finite or countable chart cover suffices. For a source with boundary apply [F4] separately on its interior and boundary, retaining the full $n$-column jet in both families; the source dimensions are $n$ and $n-1$. Since $n-(m-n+1)=2n-m-1<0$, good slices meet no rank-deficient stratum. By [F8] the union of the exceptional null sets is null. Choose a sufficiently small good parameter; immersion persists on $D_1$ by compactness and smallness. The resulting map $h$ is an immersion, unchanged near $C$. It can also be kept injective on $D_1$: local injectivity of $f$ there persists by projecting target charts to $n$ coordinates and integrating a derivative uniformly close to an invertible matrix along source-chart segments, while compactness gives a positive image separation for the remaining pairs in $D_1$.

For $n=0$, compact $N$ is finite; take $\delta >0$ below all distances between distinct source points, so the close-pair assertion is vacuous. For $n>0$, a finite collection of convex source charts gives a uniform local injectivity estimate stable under small $C^1$ perturbation of $h$: in each chart project a target chart to $n$ coordinates with derivative near a fixed invertible matrix $L$. If that derivative differs from $L$ by less than its least singular value, integrate along the segment between two source points to obtain a positive lower bound on the difference of their projected images. Compactness gives finitely many such charts and a Lebesgue radius $\delta$ for their smaller cover. Every sufficiently close map therefore separates distinct pairs of source distance less than $\delta$. Boundary half-charts are convex too.

Use finer bump charts of diameter less than $\delta \mathrm{/}3$, with independent constant-vector parameters after the same target retraction. Their profiles vanish on $D_0$ and span values outside $D_1$. Because $h$ is injective on $D_1$, compactness gives a positive image separation for pairs in $D_1$ at source distance at least $\delta \mathrm{/}2$; sufficiently small perturbations preserve it. On the open pair region of source distance greater than $\delta \mathrm{/}2$, any coincidence therefore has at least one point outside $D_1$. A value-spanning profile at that point has support excluding the other point, so the two-point evaluation is transverse to the target diagonal at every possible coincidence. Apply parametric transversality separately on the interior/boundary pair strata, each of dimension at most $2n$, and use [F8] to combine the exceptional sets. The inequality $2n-m<0$ excludes all such pairs for a sufficiently small good parameter; no smoothness of a distance-level boundary is required. Step 4.1 excludes the remaining pairs and ensures immersion persists. The resulting map is an injective immersion of a compact manifold, hence an embedding. Straight parameter segments and the retraction give a smooth homotopy to $f$, fixed near the protected set, with the initial boundary-collar homotopy included only in the boundary-only case; reparametrize at concatenation points to make the homotopy smooth. Taking both perturbations sufficiently small gives the asserted arbitrary $C^1$ closeness in the unrestricted case. All profiles and cover choices are finite; countable choice enters through the declared embedding, collar, tube and genericity suppliers.