# Tagged Questions

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0answers
47 views

### Second Countability hypothesis for a Banach manifold

Is the second countability hypothesis necessary to rigoruosly define a Banach Manifold (say in infinite dimension)? In the finite-dimensional theory of manifolds, that request is included in the ...
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### Are there analogs of smooth partitions of unity and good open covers for PL-manifolds?

Smooth partitions of unity and differentiable good open covers are important technical tools in the realm of smooth manifolds. Are there analogs of these tools for piecewise linear manifolds? A PL ...
1answer
448 views

### Do partitions of unity exist if we impose additional conditions on the derivatives?

Let $~~\cup_{k=-1}^{\infty} U_k = \mathbb{R}$ be an open covering of $\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to the cover exists, i.e. there exists smooth ...
1answer
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### How does the lack of partitions of unity affect the structure of analytic/holomorphic manifolds?

The standard way to define integration on a smooth manifold is to use partitions of unity, to extend to the case where the form you're integrating isn't supported on just one coordinate patch. Of ...
3answers
470 views

### symplectic form with partition on unity

Assume $M$ is a $2n-$dimensional differentiable manifold. Let $(U_{i})$ be a open covering of $M$. With respect to this covering let $\rho_{i}$ be a partition of unity. Assume that on each $U_{i}$ we ...
3answers
2k views

### What is the right version of “partitions of unity implies vanishing sheaf cohomology”

There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...
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### An “existence contra partition of unity” statement for integer matrices?

While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind. Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...