All Questions
Tagged with dg.differential-geometry at.algebraic-topology
639 questions
14
votes
3
answers
2k
views
Recommendations for getting into sheaves with emphasis on differential geometry and algebraic topology
I want to study the theory of sheaves from a categorical point of view with an emphasis on applications in algebraic topology and differential geometry and I'm looking for a good introductory book to ...
- 141
14
votes
1
answer
681
views
When does an open manifold admit two linearly independent vector fields?
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$\DeclareMathOperator{\co}{H}$
$\newcommand{\kk}{\mathbb{F}}$
$\newcommand{\qq}{\mathbb{Q}}$
$\newcommand{\zz}{\mathbb{Z}}$
$\newcommand{\rr}{\mathbb{R}}$
$\...
- 1,726
14
votes
1
answer
480
views
"Small" maps from sphere to sphere
Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co-dimension $n$). Now ...
- 17.6k
14
votes
1
answer
2k
views
what is the universal cover of GL(2,R)?
In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.
What is G'?
I know there is concrete description in terms of pairs (M,f) ...
- 303
14
votes
1
answer
573
views
Different proof techniques of the Atiyah-Singer index theorem
I am aware of the usual K-theoretical (cobordism, operator algebras) and heat kernel proofs of the index theorem, as answered in other questions in this site, e.g. here.
However, I recently read this ...
14
votes
3
answers
3k
views
Errata for Bott and Tu's book "Differential Forms in Algebraic Topology"
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Tu is a prequel.
Is there a good list of errata for Bott and Tu available? ...
- 315
14
votes
0
answers
573
views
Reference for a proof of the fiberwise Stokes theorem
The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary,
the difference between the fiberwise integral of the differential and the ...
- 37.8k
13
votes
4
answers
3k
views
singular homology of a differential manifold
Let $M$ be a differentiable manifold, $\Delta$ the closed simplex $[p_0, p_1,...,p_k]$. A differential singular $k$-simplex $\sigma$ of $M$ is a smooth mapping $\sigma:\Delta \to M$.
And we construct ...
- 750
13
votes
4
answers
2k
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Fundamental groups of compact Kähler manifolds
This is a sort of a follow-up to this question, and especially to Sean Lawton's answer: The book Fundamental Groups of compact Kähler manifolds (which, in my opinion, is one of the best mathematics ...
- 96.4k
13
votes
2
answers
2k
views
Given a complex vector bundle with rank higher than 1, is there always a line bundle embedded in it?
When I read the GTM 082, "The Splitting Principle of the complex vector bundle", I see that in the proof we split off one subbundle at a time by pulling back to the projectivization of a ...
- 157
13
votes
3
answers
1k
views
Manifold whose universal covering is a sphere but which is not a space form?
Let $M^n$ be a smooth manifold whose universal cover is homeomorphic $\mathbb{S}^n$, are there examples where $M^n$ is not homeomorphic to a space form ?
The answer may vary if you replace ...
- 4,123
13
votes
3
answers
851
views
Are negatively pinched manifold locally conformally flat?
One knows that hyperbolic manifolds are locally conformally flat.
How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy:
$$
-\Lambda \le K \le -\lambda$$
for $\Lambda&...
- 2,623
13
votes
4
answers
4k
views
Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds
I would like to find a pedagogical reference where the classification, up to isomorphism, of principal $SU(2)$ bundles over a four-dimensional compact, oriented manifold is explained. In particular I ...
- 2,816
13
votes
4
answers
3k
views
Circle bundles over $RP^2$
Does anybody know if orientable, closed $3$-manifolds that are circle bundles over $RP^2$ have been classified?
One can determine the isomorphism classes of bundles using obstruction theory, but I am ...
13
votes
2
answers
700
views
Are manifolds admitting a circle foliation covered by manifolds with a (non-trivial) circle action?
More precisely, is there a criterion that decides the above question?
I am particularly interested in the smooth setting: is a smooth manifold with a smooth regular foliation by circles covered by a ...
- 183
13
votes
2
answers
1k
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Realizing cohomology classes by submanifolds
In "Quelques propriétés globales des variétés différentiables", Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. ...
- 5,824
13
votes
2
answers
2k
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Nice example of a topologically trivial bundle with nontrivial connection
So, I've been trying to understand what exactly an anomaly is, and how they arise in physics. Apparently an anomalous theory is some theory whose action is given by a section of some bundle (rather ...
13
votes
3
answers
707
views
Can a homotopy inverse of the map from a Lie group to loops on its classifying space be given by holonomy?
Let $G$ be the compact Lie group $SO(n)$. There are some classical constructions of the classifying bundle of $G$ based upon on direct limits of Grassmann and Stiefel manifolds:
$$BG \simeq \...
- 53.3k
13
votes
1
answer
1k
views
Is every orientable circle bundle principal?
The only examples I found of nonprincipal circle bundle are nonorientable, like the Klein bottle that is an S^1 bundle over S^1 which is not principal and nontrivial. That makes me ask the question.
...
- 133
13
votes
1
answer
2k
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de rham model for relative cohomology
In GTM82, I read a model for the relative cohomology of (M,N) with N a submanifold of M.
And in the page:
Relative De Rham cohomologies,
I got to know that there is another model for relative ...
- 303
13
votes
1
answer
731
views
free loop space and invariant forms
Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the ...
- 2,035
13
votes
1
answer
637
views
Can a PDE constrain the degree of a $C^\infty$ map germ?
Let $\pi:E\to M$ be a smooth vector bundle over a smooth manifold, with $\text{rank}(E)=\text{dim}(M)$. For a section $\sigma$ of $E$ with a zero at $p\in M$, define the degree of the zero at $p$ to ...
- 3,212
13
votes
0
answers
388
views
Does the existence of an almost complex structure solely depend on the topology of the manifold?
To be precise, let $M$ and $N$ be two 2n-dimensional smooth, closed manifolds that are homeomorphic. If $M$ admits an almost complex structure, can we deduce that $N$ also admits an almost complex ...
- 661
13
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0
answers
319
views
Exotic smooth structures on Fano manifolds
If two Fano projective manifolds are homeomorphic are they diffeomorphic?
There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of ...
user164740
12
votes
2
answers
767
views
Unique almost complex structure up to diffeomorphism
For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?
For example it is true for $S^2$.
user164740
12
votes
3
answers
3k
views
Are the stiefel-Whitney classes of the tangent bundle determined by the mod 2 cohomology?
Let $G=\mathbb{Z}/2\mathbb{Z}$. Let $f\colon L \to N$ be a smooth map of connected smooth closed $n$-dimensional manifolds such that the induced map
$$f^* \colon H^*(N,G) \to H^*(L,G)$$
is an ...
- 2,590
12
votes
3
answers
860
views
A nontrivial principal bundle which satisfies Leray-Hirsch theorem
What is an example of a nontrivial principal bundle whose fibre space $G$, total space $P$ and base space $M$ are compact connected manifolds (the fiber $G$ is a compact Lie group) such that $$H^*(P,\...
- 356
12
votes
2
answers
887
views
Representation viewpoint on Chern–Weil (cohomology computations done with rep theory?)
$\DeclareMathOperator\Sym{Sym}$Let $G$ be a compact lie group. Chern–Weil theory tells us that there's a homomorphism:
$$H^{*}(BG;\mathbb{R}) \to (\Sym^{\bullet} \mathfrak{g^*})^G$$
which in our case ...
- 7,789
12
votes
2
answers
597
views
Steenrod powers of Pontryagin classes
It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true ...
- 1,452
12
votes
1
answer
1k
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Geometry of Whitehead manifolds.
I'm currently studying some problems about the Whitehead manifold $W$ (the open 3-manifold which is contractible but not homeomorphic to $\mathbb{R}^3$). Does there exists some survey paper on its ...
- 4,123
12
votes
2
answers
757
views
Finding topological obstructions for a complex manifold to be Kaehler
Well, it is of the "straightforward" questions one may ask. I propose it here to see if someone could tell me more on the recent status of this quite long-standing problem.
To initiate, let me give a ...
- 355
12
votes
2
answers
693
views
Cohomology of the quotient of a Lie group by a finite subgroup
Let $G$ denote the $\operatorname{Spin}(n)$ group with $n>4$ and let $\Gamma$ be a cyclic subgroup $G$ of a prime order $p >2$. When does the projection $G \to G/\Gamma$ induce a surjection
...
12
votes
1
answer
840
views
Reference request: Topology on the space of smooth compact submanifolds
In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ ...
- 121
12
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3
answers
1k
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A version of Lusternik–Schnirelmann category for good open covers
Recall that the Lusternik–Schnirelmann category (or LS-category) of a space is the integer $n$ such that there is an open cover by $n+1$ open sets which have nullhomotopic inclusions, and no such ...
- 35.5k
12
votes
1
answer
3k
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Poincaré duality with boundary conditions
If $M$ is a compact oriented manifold with boundary then by Poincaré duality the cohomology of $\Omega(M)$ (de Rham cohomology of $M$) is dual to the cohomology of $\Omega_0(M)$, where $\Omega_0(M)$ ...
- 407
12
votes
1
answer
896
views
Analytic Torsion in the Derived Category
I recently learned about analytic torsion and about the amazing Cheeger-Muller theorem identifying analytic and Reidemeister torsion for compact Riemannian manifolds.
Now analytic torsion is defined ...
- 23k
12
votes
1
answer
482
views
Characterize spin cobordism invariants in dimer models
The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...
- 875
12
votes
0
answers
1k
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Are there exotic $S^2\times S^2$?
On 2010 AKHMEDOV and PARK claimed there are infinitely many exotic smooth structures on $S^2\times S^2$, see http://arxiv.org/abs/1005.3346
Then Rasmussen posted a paper : Perfect Morse functions and ...
- 2,623
11
votes
1
answer
620
views
Is $SL(n,\mathbb{Z})$ a CAT(0) group?
Is it possible to find a CAT(0) space on which the matrix group $SL(n,\mathbb{Z})$ acts properly discontinuously and cocompactly? Note: when the cocompactness is dropped , it is possible.
- 1,373
11
votes
2
answers
1k
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Two approaches to compute the signature of a Kaehler manifold
Given a compact Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H_{2n}(M,\mathbb{R})$:
...
11
votes
2
answers
423
views
Does combinatorial formula for the Pontrjagin classes exist?
Gelʹfand, I. M. and MacPherson, R. D. "A combinatorial formula for the Pontrjagin classes" Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 304–309.
In the above paper the authors claimed a ...
- 2,623
11
votes
1
answer
723
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representatives of the group of homotopy 7-spheres
In Milnor's paper "On manifolds homeomorphic to the 7-sphere" it is proven that there are manifolds homeomorphic but not diffeomorphic to the standard 7-sphere. His construction involves sphere ...
- 1,415
11
votes
2
answers
811
views
Higher dimensional Heegaard splittings?
Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing ...
- 732
11
votes
1
answer
579
views
Fourth obstruction, Pontryagin and Euler class
Assume the first three obstruction classes of a rank 4 vector bundle vanish and look at the fourth obstruction class. This fourth obstruction class can be decomposed as the Euler class and the first ...
- 4,432
11
votes
1
answer
2k
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A survey for various $K$-homology theories and their relationship
The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology theory....
- 593
11
votes
2
answers
987
views
first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles
Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to $SO(...
- 2,223
11
votes
1
answer
1k
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Pontrjagin numbers and exotic spheres
Hi everyone, im reading Milnor's article "On manifolds homeomorphic to the 7-sphere", in which he constructs the first example of an exotic structure, id like to know if there's a particular reason ...
- 301
11
votes
1
answer
548
views
Characterizing flat 2-connections by their holonomy
Hello,
A flat principal $G$-bundle over $X$ is determined by its holonomies, which are (after picking a trivialization) group homomorphisms $\pi_1(X)\rightarrow G$. The fiber of the bundle is not ...
- 543
11
votes
1
answer
593
views
Examples of 6-manifolds without an almost complex structure
Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the ...
- 6,995
11
votes
1
answer
379
views
Smooth structure on direct product
Let $M$ be the $E_8$ manifold. Is there a closed manifold $N$ such that $M\times N$ is smoothable? What is the smallest possible dimension of $N$?
user164740