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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.

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Paracompactness for a smooth space is equivalent to M admitting a smooth partition of unity, or that every open covering has a smooth partition of unity subordinate to it? – Mariano Suárez-Alvarez Jan 19 '12 at 8:18
Thx- corrected. – Daniel Moskovich Jan 19 '12 at 8:20
If you have two smooth extensions $F_0$ and $F_1$ of $f$, then $F_t=(1-t)F_0+tF_1$ is a smooth extension of $f$. – Ben McKay Jan 19 '12 at 8:25
But on the other hand, there are many smooth extensions which are not equivalent under diffeomorphisms; just arrange different numbers of critical points away from the set $A$. – Ben McKay Jan 19 '12 at 8:27
up vote 2 down vote accepted

Typically, partitions of unity are used to prove a statement along the following lines. Given a paracompact $X$ and for each open set $U\subset U$ a certain space $S_U$ which satisfies an appropriate sheaf condition ($U \mapsto S_U$ is a sheaf of spaces). You want to prove that $S_X$ is nonempty. Often, a small improvement of the argument gives the better conclusion that $S_X$ is contractible.

Examples: $E \to X$ a vector bundle, $S_U$ the space of bundle metrics on $E|_U$. For trivial reasons, $S_U$ is always convex. The partition of unity argument shows that $S_X$ is nonempty (because each point of $X$ has a neighborhood $U$ $S_U$ is nonempty; use the partition of unity to glue things together). Therefore, the space of metric is contractible; a metric constructed using partitions of unity is unique up to ''isotopy'' in the appropriate sense. Because of the convexity, partitions of unity do not play a role for the proof of contractibility.

Other examples along the same lines are connections, differential operators with a given principal symbol or the extensions of a given smooth function. In these cases it is pretty clear that the global objects you construct belong to a convex space.

There are more subtle examples, where the spaces $S_U$ do not have linear structure. For example, if $P \to X$ is a $G$-principal bundle and you want to construct a $G$-equivariant map $P \to EG$, a contractible free $G$ space. If $P$ is trivial, the space of these maps is contractible. One uses partitions of unity to glue these local data together. The same argument almost shows that the space $map_G (P;EG)$ has trivial homotopy groups (extend equivariant maps given on a nicely embedded subspace $Y \subset X$). The chapter ''partitions of unity in homotopy theory'' of tom Diecks book has a lot more of this type.

Sometimes, you really want to know that the space of partitions of unity is itself contractible. This does not make sense without further clarifications what the space of partitions of unity should be. Here is one possibility to do it. Let $X$ be a manifold and $I$ a fixed infinite set. Make a simplicial set whose $p$-simplices are tuples $(U_i,\lambda_i)_{i \in I}$, where $U_i$ is an open cover of $M \times \Delta^p$ and $\lambda_i$ is a subordinate locally finite partition of unity. This is a contractible simplicial set.

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Thanks- this is a really helpful answer. – Daniel Moskovich Jan 20 '12 at 2:07

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