One of the requirements for a smooth manifold $M$ is that it be paracompact, and one of the equivalent definitions of paracompactness for a smooth space is that for overy open cover of $M$, there exists a smooth partition of unity subordinate to it. Partitions of unity allow us to glue and to extend diffeomorphisms- see e.g this question. More fundamentally perhaps, there is the Steenrod Approximation Theorem, which gives (under mild conditions) that an extension $f$ of a smooth function on a closed subset $A\subset M$ by a continuous function may be improved to an extension $F$ by a smooth function on $M$.

The approximation $F$, like all gluings and extensions in differential topology, depends on the smooth partition of unity used to construct it. So it seems obvious that $F$ can't possibly be unique. Is it at least unique up to isotopy? I.e., for two such approximations $F_{0,1}$, is there a smooth structure on $M\times I$ to the target cross $I$ with a smooth $1$-parameter family of smooth maps $F_{(t)}$ from $M\times {\{t\}}$, and $F_{(0)}=F_0$ and $F_{(1)}=F_{1}$?. More generally:

Question: Is a smooth map which is constructed using a smooth partition of unity (as an extension of a smooth map, or as an approximation of a diffeomorphism) unique up to isotopy or only up to diffeomorphism? Is there a reference?

I'm confused about this point and the issues which surround it. I suppose that a related question is, if one looks at the subspace of smooth functions made up of partitions of unity, with the induced $C^\infty$ topology, is it connected? Is it simply-connected? What about higher homotopy groups? Are there references which discuss these types of issues?

When thinking about categories of cobordisms, one really wants gluings to be unique up to isotopy, but classical texts (Hirsch, Munkres, etc.) only claim uniqueness up to diffeomorphism, and to "improve" their results (which I'm sure many people have done before), this question seems to inevitably arise.

One of the requirements for a smooth manifold $M$ is that it be paracompact, and one of the equivalent definitions of paracompactness for a smooth space is that for overy open cover of $M$, there exists a smooth partition of unity subordinate to it. Partitions of unity allow us to glue and to extend diffeomorphisms- see e.g this question. More fundamentally perhaps, there is the Steenrod Approximation Theorem, which gives (under mild conditions) that an extension $f$ of a smooth function on a closed subset $A\subset M$ by a continuous function may be improved to an extension $F$ by a smooth function on $M$.

The approximation $F$, like all gluings and extensions in differential topology, depends on the smooth partition of unity used to construct it. So it seems obvious that $F$ can't possibly be unique. Is it at least unique up to isotopy? I.e., for two such approximations $F_{0,1}$, is there a smooth structure on $M\times I$ to the target cross $I$ with a smooth $1$-parameter family of smooth maps $F_{(t)}$ from $M\times {\{t\}}$, and $F_{(0)}=F_0$ and $F_{(1)}=F_{1}$?. More generally:

Question: Is a smooth map which is constructed using a smooth partition of unity (as an extension of a smooth map, or as an approximation of a diffeomorphism) unique up to isotopy or only up to diffeomorphism? Is there a reference?

I'm confused about this point and the issues which surround it. I suppose that a related question is, if one looks at the subspace of smooth functions made up of partitions of unity, with the induced $C^\infty$ topology, is it connected? Is it simply-connected? What about higher homotopy groups? Are there references which discuss these types of issues?

When thinking about categories of cobordisms, one really wants gluings to be unique up to isotopy, but classical texts (Hirsch, Munkres, etc.) only claim uniqueness up to diffeomorphism, and to "improve" their results (which I'm sure many people have done before), this question seems to inevitably arise.

One of the requirements for a smooth manifold $M$ is that it be paracompact, and one of the equivalent definitions of paracompactness for a smooth space is that $M$ admits a smooth partition of unity. Partitions of unity allow us to glue and to extend diffeomorphisms- see e.g this question. More fundamentally perhaps, there is the Steenrod Approximation Theorem, which gives (under mild conditions) that an extension $f$ of a smooth function on a closed subset $A\subset M$ by a continuous function may be improved to an extension $F$ by a smooth function on $M$.

The approximation $F$, like all gluings and extensions in differential topology, depends on the smooth partition of unity used to construct it. So it seems obvious that $F$ can't possibly be unique. Is it at least unique up to isotopy? I.e., for two such approximations $F_{0,1}$, is there a smooth structure on $M\times I$ to the target cross $I$ with smooth maps $F_{(t)}$ from $M\times {\{t\}}$, and $F_{(0)}=F_0$ and $F_{(1)}=F_{1}$?. More generally:

Is a smooth map which is constructed using a smooth partition of unity (as an extension of a smooth map, or as an approximation of a diffeomorphism) unique up to isotopy or only up to diffeomorphism? Is there a reference?

I'm confused about this point and the issues which surround it. I suppose that a related question is, if one looks at the subspace of smooth functions made up of partitions of unity, with the induced $C^\infty$ topology, is it connected? Is it simply-connected? What about higher homotopy groups? Are there references which discuss these types of issues?

When thinking about categories of cobordisms, one really wants gluings to be unique up to isotopy, but classical texts (Hirsch, Munkres, etc.) only claim uniqueness up to diffeomorphism, and to "improve" their results (which I'm sure many people have done before), this question seems to inevitably arise.

One of the requirements for a smooth manifold $M$ is that it be paracompact, and one of the equivalent definitions of paracompactness for a smooth space is that for overy open cover of $M$, there exists a smooth partition of unity subordinate to it. Partitions of unity allow us to glue and to extend diffeomorphisms- see e.g this question. More fundamentally perhaps, there is the Steenrod Approximation Theorem, which gives (under mild conditions) that an extension $f$ of a smooth function on a closed subset $A\subset M$ by a continuous function may be improved to an extension $F$ by a smooth function on $M$.

The approximation $F$, like all gluings and extensions in differential topology, depends on the smooth partition of unity used to construct it. So it seems obvious that $F$ can't possibly be unique. Is it at least unique up to isotopy? I.e., for two such approximations $F_{0,1}$, is there a smooth structure on $M\times I$ to the target cross $I$ with a smooth $1$-parameter family of smooth maps $F_{(t)}$ from $M\times {\{t\}}$, and $F_{(0)}=F_0$ and $F_{(1)}=F_{1}$?. More generally:

Question: Is a smooth map which is constructed using a smooth partition of unity (as an extension of a smooth map, or as an approximation of a diffeomorphism) unique up to isotopy or only up to diffeomorphism? Is there a reference?

I'm confused about this point and the issues which surround it. I suppose that a related question is, if one looks at the subspace of smooth functions made up of partitions of unity, with the induced $C^\infty$ topology, is it connected? Is it simply-connected? What about higher homotopy groups? Are there references which discuss these types of issues?

When thinking about categories of cobordisms, one really wants gluings to be unique up to isotopy, but classical texts (Hirsch, Munkres, etc.) only claim uniqueness up to diffeomorphism, and to "improve" their results (which I'm sure many people have done before), this question seems to inevitably arise.