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Results tagged with manifolds
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user 402
A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.
16
votes
Accepted
Manifolds with negative dimension – Definition, References
Smooth manifolds of negative dimension are defined in derived geometry. … See Spivak, Derived Smooth Manifolds. …
- 37.8k
9
votes
Accepted
To what extent differentiable mappings of an affine line into a manifold determine its diffe...
This is true for infinitely differentiable curves: if a map sends smooth curves to smooth curves, then it is smooth, by a theorem of Boman from 1967:
Jan Boman. Differentiability of a Function and of …
- 37.8k
9
votes
1
answer
401
views
Reference for the Brown-Gersten property for smooth manifolds
There is an analog of this result for smooth manifolds: the homotopy descent property
for the open cover topology on smooth manifolds boils down to the descent for Mayer-Vietoris squares
and descent for … For the case of smooth manifolds, a recent paper by Kreck and Singhof “Homology and cohomology theories on manifolds”
has an explicit formulation of the axioms of cohomology theories on smooth manifolds …
- 37.8k
2
votes
Accepted
Reference for the Brown-Gersten property for smooth manifolds
I typed up a proof of this result:
Numerable open covers and representability of topological stacks.
The result is proved in greater generaility for arbitrary numerable open covers of topological sp …
- 37.8k
4
votes
(Homotopy) colimit and manifold
So my first question is, can a “diagram of spaces over a CW complex” be a manifold? If so, under what conditions?
Any smooth manifold is homotopy equivalent to the homotopy colimit of a diagram of c …
- 37.8k
6
votes
Accepted
What are some "good" examples of Kan simplicial manifolds?
Kan simplicial manifolds are in the same relation to differentiable ∞-stacks
(i.e., locally fibrant simplicial presheaves on the site of cartesian spaces and smooth maps)
as smooth manifolds are to sheaves … That is to say, Kan simplicial manifolds can be seen as the ∞-categorification of manifolds.
Some important examples include:
Any ordinary manifold, interpreted as a constant simplicial object. …
- 37.8k
5
votes
Accepted
Metrics on derived smooth manifolds
As far as I am aware, there is nothing in the literature that
treats Riemannian or pseudo-Riemannian metrics on derived smooth manifolds. …
- 37.8k
6
votes
0
answers
392
views
Do all topological manifolds admit locally flat embeddings into R^n?
In his 1969 paper “Locally flat imbeddings of
topological manifolds” Lees proved that a closed oriented
second countable topological manifold admits a locally flat embedding into some R^n. …
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2
votes
0
answers
237
views
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? … Do PL-manifolds admit PL good open covers? Do open covers of PL-manifolds admit subordinate PL partitions of unity? …
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3
votes
A k-form is thought of as measuring the flux through an infinitesimal k-parallelepiped
See Theorem 1 in Anders Kock's paper
“Differential forms as infinitesimal cochains”,
which is devoted precisely to this question.
Specifically, the map b in the formula (1)
establishes an explicit bij …
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