The manifolds tag has no wiki summary.

**18**

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**3**answers

2k views

### A book on locally ringed spaces?

Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a ...

**10**

votes

**2**answers

749 views

### Meaning of orientation/orientability over rings other than the integers

This was asked as part of an earlier question. But since this part did not attract many answers, I am asking it separately.
We consider the homology definition of an orientation for a manifold, as ...

**4**

votes

**2**answers

416 views

### Help me understand boundary terms in actions over nontrivial manifolds

So I have this manifold $M$, along with a metric $g_{\mu\nu}(x)$ and metric-compatible covariant derivative $\nabla_\mu$ (which is not necessarily the one corresponding to the Levi--Civita ...

**16**

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**15**answers

3k views

### Important results that use infinite-dimensional manifolds?

Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn up? Important ...

**2**

votes

**1**answer

404 views

### Special map from a manifold to GL_n(R)?

I've just finished my first course in differential geometry, so forgive me if this is maybe a silly or well-known question, but given any, say, diffeomorphism of $n$-manifolds $\phi:M\rightarrow N$, I ...

**21**

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**3**answers

1k views

### What manifolds are bounded by RP^odd?

Real projective spaces ℝPn have ℤ/2 cohomology rings ℤ/2[x]/(xn+1) and total Stiefel-Whitney class (1+x)n+1 which is 1 when n is odd, so it follows that odd dimensional ones are boundaries of compact ...

**15**

votes

**3**answers

1k views

### Algebraic varieties which are topological manifolds

Inspired by this thread, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space ...

**6**

votes

**3**answers

539 views

### Maximal exotic $\mathbb{R}^4$

Article Exotic $\mathbb{R}^4$ on Wikipedia says that there is at least one maximal smooth structure on $\mathbb{R}^4$, that is such an atlas on $\mathbb{R}^4$ that any other smooth $\mathbb{R}^4$ can ...

**5**

votes

**6**answers

4k views

### What books should I read before beginning Masaki Kashiwara's “Sheaves on Manifolds”

I am a beginner trying to learn about sheaves. I am trying to read Masaki Kashiwara's book "Sheaves on Manifolds", but I find it is not easy for me to understand.
What other books should I read ...

**8**

votes

**5**answers

778 views

### Simplicial volume

Is there a finite dimensional closed manifold $M$ which is a $K(\pi,1)$, whose fundamental group is not word-hyperbolic, but which has a positive simplicial volume (ie "Gromov norm")?
(Added:) The ...