Encounters with partitions of unity

Question

Not sure how this would be received here. This question is about smooth partitions of unity.

Let $M$ be a manifold. Consider an open cover $\{U_\alpha\}_{\alpha\in \Lambda}$ of $M$. A collection of smooth functions$\{p_\alpha:U_\alpha\rightarrow \mathbb{R}\}_{\alpha\in \Lambda}$ is called a smooth partition of unity subordinate to the cover $\{U_\alpha\}$ if

1. $\text{supp}(p_\alpha)\subseteq U_\alpha$ for each $\alpha\in \Lambda$,
2. the collection of supports $\{\text{supp}(p_\alpha)\}_{\alpha\in \Lambda}$ is a locally finite set,
3. $\sum_{\alpha\in \Lambda}p_\alpha=1$.

It is known that given any open cover $\{U_\alpha\}$, we can produce a partition of unity on $M$ subordiante to this cover. Next question is, what is the use of partitions?

1. Suppose I am given a $n$-form $\omega$ on a (oriented) $n$-manifold $M$. I have to make sense of (give a reasonable definition for) $\int_M\omega$. Suppose that $\omega$ is (compactly supported) zero outside a chart $(U,\phi:U\rightarrow \mathbb{R}^n)$ (let $\psi:\phi(U)\subseteq \mathbb{R}^n\rightarrow U$ be its inverse). Then, pullback $\omega$ along $\psi$ to get $n$-form $\psi^*\omega$ on $\phi(U)\subseteq \mathbb{R}^n$ and we know how to integrate an $n$-form on an open subset of $\mathbb{R}^n$. So, we know what is $\int_{\phi(U)}\psi^*\omega$ is and define $\int_M\omega:=\int_{\phi(U)}\psi^*\omega$. Suppose $\omega$ is arbitrary (compactly supported), then, ask for partition of unity $\{p_\alpha\}$ and consider $p_\alpha\omega$. This is compactly supported in $U_\alpha$. So, $\int_{U_\alpha}p_\alpha\omega$ make sense and define $\int_{M}\omega=\sum \int_{U_\alpha}p_\alpha\omega$.
2. Given a manifold $M$, how do I know that there exists a Riemannian metric on $M$? Partition of Unity.
3. Given a manifold $M$, how do I know there exists a connection on the tangent bundle $TM\rightarrow M$? Partitions of unity.
4. Given a principal bundle over manifold $M$, how do I know connection exists on the principal bundle $P\rightarrow M$? Partitions of unity (along with trivialization of course).

My question is the following:

Is partition of unity used for anything serious than making sure some structures can be glued to give a global structure? Do you, as a research mathematician, come across the necessity of using the partition of unity for any reason other than similar to what I mentioned above?

Answer 1

How do you know a bundle has a classifying map? Partition of unity. This is a pretty big deal, since a bundle without a classifying map doesn't inherit structure from the universal bundle.

Partitions of unity are also important in homotopy theory, check out Dold's original paper

Albrecht Dold, Partitions of unity in the theory of fibrations, Ann. of Math. 78 (1963) pp 223–255, doi:10.2307/1970341 (Rainicki's archive)

as well as

Tammo tom Dieck, Partitions of unity in homotopy theory, Compositio Mathematica 23 (1971) no. 2, pp 159–167 (NUMDAM)

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Added: one big important result is that on a paracompact Hausdorff space $X$ (where every open cover admits a subordinate partition of unity) Čech cohomology calculates sheaf cohomology, which is the "real" cohomology of $X$.

Answer 2

Partition of unity allows one to prove that the sheaf of smooth functions is acyclic - which is needed in the construction of de Rham cohomology.

As such, it is also completely analogous to the (weak) Hilbert's Nullstelensatz in algebraic geometry, which allows to break 1 into sum of polynomials vanishing in some closed subsets provided the intersection of these subsets is 0.

Answer 3

Partitions of unity play a key role in Weil’s triangulation-free proof of de Rham’s theorems (1952; pdf).

Chorlay (2010; pdf) has some interesting history of the concept and its use in other proofs (Whitney, Bochner, Weyl, L. Schwartz, Cartan 1951, Serre 1952, ...).