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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?

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    $\begingroup$ I only use partitions of unity to glue. $\endgroup$ – Ben McKay Oct 8 '19 at 10:39
  • $\begingroup$ @BenMcKay I also use (very rarely) to glue.. I thought it can be used for something more... $\endgroup$ – Praphulla Koushik Oct 8 '19 at 10:51
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    $\begingroup$ Reason for downvote? I will try to make it better if I am told what is the problem with the question... $\endgroup$ – Praphulla Koushik Oct 8 '19 at 12:53
  • $\begingroup$ Your phrasing "anything serious than ..." sounds as if the possibilty of gluing would be a minor issue. To me, gluing seems to be a basic, but very important concept. You can glue arbitrary Riemannian metrics, so they always exist. But you cannot glue flat Riemannian metrics, so you have to work hard to produce them (and they don't exist on most manifolds ...). The question whether gluing is possible for a particular problem leads to some interesting mathematics. I suspect, $h$-principles were inspired by similar questions ... (and no, I did not downvote) $\endgroup$ – Sebastian Goette Oct 10 '19 at 14:08
  • $\begingroup$ @SebastianGoette thank you for your response.. I did not mean it is a minor issue :) though I am yet to fully realise its seriousness, I only realise it partially as of now.. I am trying to understand what h principle is doing here.. I will try to google and ask you if I don’t get anything... $\endgroup$ – Praphulla Koushik Oct 10 '19 at 17:19
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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)


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 $X$.

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  • $\begingroup$ Thank you.. I will see those and ask if I have any specific question.. When you have some free time, please write about how differently or similarly does partition of unity come in homotopy theory as mentioned in above papers as compared to partitions of unity in smooth manifolds.. I will also read and write what I understand (hopefully with in 10 days).... $\endgroup$ – Praphulla Koushik Oct 8 '19 at 12:25
  • $\begingroup$ The main thing I think is that there is a constraint in that the existence of a partition of unity subordinate to any given cover of a T1 space $X$ is equivalent it to being paracompact and Hausdorff $\endgroup$ – David Roberts Oct 8 '19 at 22:22
  • $\begingroup$ Oh. Ok ok. Thank you. I will read that articles and respond.. $\endgroup$ – Praphulla Koushik Oct 11 '19 at 18:53
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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.

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  • $\begingroup$ I did not look at sheaf version of deRham cohomology construction seriously.... now there is enough motivation to do that. Thank you... $\endgroup$ – Praphulla Koushik Oct 9 '19 at 2:58
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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, ...).

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  • $\begingroup$ Thank you, there are some other articles in that web page of Chorlay... they seem to be interesting... thanks again.. $\endgroup$ – Praphulla Koushik Oct 9 '19 at 2:59

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