All Questions
Tagged with dg.differential-geometry at.algebraic-topology
639 questions
5
votes
2
answers
2k
views
Canonical reference for Chern characteristic classes
I'm a little uncertain about the definitions for
Chern roots
Chern classes
Chern characters
From perusing several discussions, I gather that if one correlates the nomenclature with that of ...
- 10.5k
2
votes
0
answers
243
views
What does one call a Morse function whose nondegenerate condition is relaxed?
In robotics, navigation functions are of utmost interest to plan a path from an initial location $q_0$ to a target location $q^t$. A function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a navigation ...
user106458
6
votes
0
answers
230
views
Equivariant Venice Lemma
In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as
Theorem: For ...
- 386
6
votes
0
answers
132
views
On the weak homotopy type of a differentiable (Chen) space
Suppose that $M$ is a differentiable space in the sense of Chen (cf. https://ncatlab.org/nlab/show/Chen+space ).
Assume that $M$ also has the structure of a topological space and that the two ...
- 18.8k
5
votes
1
answer
366
views
K-theory for a (geometric) stack
There is a notion of $K$-theory for a manifold $M$.
Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...
- 6,023
2
votes
2
answers
214
views
Measuring failure of a setup to preserve some structure giving interesting notions
I am looking for some examples of failure of some structures giving interesting notions. For example, we have the following situation:
Let $P(M,G)$ be a principal bundle. Let $\Gamma\subseteq TP$ be ...
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
1
vote
0
answers
300
views
Constructions that can be seen as objects representing a functor
Some constructions can be seen as objects representing a functor.
For example,
Consider a topological group $G$ and a functor $\mathcal{F}:\text{Top}\rightarrow \text{Gpd}$ defined as $M\mapsto \...
- 6,023
5
votes
1
answer
440
views
Using Stiefel-Whitney class to build new principal bundles
I'm reading this paper and at the beginning of the second section, he states many results that aren't clear to me.
Consider a principal $SO(3)$-bundle $P\rightarrow R^2\times \Sigma$, where $\Sigma$ ...
- 311
3
votes
0
answers
127
views
Methods for constructing or checking for nontrivial classes in de Rham cohomology with local coefficients
Let $M$ be a smooth manifold (possibly with boundary), $E \to M$ a flat vector bundle, and $\mathcal{L}$ the corresponding sheaf of parallel sections.
Given a de Rham cohomology class $[\omega] \in H^...
- 6,025
7
votes
0
answers
282
views
A cohomology associated to a vector field on a Riemannian manifold
Edit: Accoring to the comment of Asura Path I revise the question.
Let $X$ be a vector field on a Riemannian manifold $(M,g)$. So we have a $1$-form $\beta$ with $\beta(Y)=\langle Y,X\...
- 356
3
votes
1
answer
985
views
Closed Poincaré dual, why $\int_M \omega \wedge \eta_S$ and not $\int_M \eta_S \wedge \omega $?
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel.
The characterization of the closed Poincaré dual ...
- 315
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
13
votes
2
answers
1k
views
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
1
vote
0
answers
150
views
Nontrivial Gauss-Manin connection
Suppose $p: X \rightarrow S$ is a fiber bundle of smooth manifold, if the Gauss-Manin connection is nontrivial, could $p$ be trivial bundle as smooth manifold? Also, could $p$ be trivial bundle as ...
- 677
6
votes
2
answers
857
views
Lifting sections of a projective bundle to a vector bundle
Let $E\to M$ be a smooth $\mathbb{K} = \mathbb{R}, \mathbb{C}$ - vector bundle over a possibly non-compact connected manifold $M$. Denote by $\mathbb{P}(E) \to M$ its projectivization, which is ...
- 2,816
4
votes
1
answer
757
views
Homotopy groups of fiber products
Let $X, Y, B$ be three smooth manifolds, and $f : X\to B$, $g : Y\to B$ submersions.
Then $X\times_BY$ exists.
(1) If $X, Y, B$ have the homotopy type of a finite CW complex, does $X\times_BY$?
(2) ...
- 180
1
vote
0
answers
57
views
$\omega$-nilpotent cover of a recurrent surface
Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville.
$\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to ...
- 391
15
votes
3
answers
2k
views
Examples of odd-dimensional manifolds that do not admit contact structure
I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
- 2,533
8
votes
1
answer
388
views
On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes
In
The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513
Woodward proposed a classification of $\mathrm{PU}...
- 81
4
votes
1
answer
198
views
Space of non-vanshing sections path-connected?
Let $M$ be a path connected smooth manifold and $E$ be a vectorbundle over $M$ of rank at least two. My question is: Under which conditions is the space of global non-vanishing sections path connected?...
- 271
14
votes
2
answers
4k
views
Mistakes in Bredon's book "Topology and Geometry"?
I am preparing the notes for a course in Algebraic Topology, so I decided to borrow some of the material from the classical (and wonderful) book by G. Bredon Topology and Geometry.
Looking at the ...
- 66.3k
7
votes
2
answers
2k
views
Is there a theorem showing that de Rham homology is isomorphic to singular homology?
The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on String Phenomenology. It was brief and gave only basic outline of how to construct this homology.
Now de ...
- 2,369
11
votes
1
answer
866
views
Serre spectral sequence for de Rham cohomology
Suppose we a given a fibration of manifolds $p\colon E\to M$ with a path connected fiber $F$ and simply connected $M$, then we have the Serre spectral sequence with
$$
E_2^{p,q} = H^p(M,\underline{H^...
- 2,305
5
votes
1
answer
379
views
Conversion formula between "generalized" Stiefel-Whitney class of real vector bundles: O(n) and SO(n)
$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$,
$$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$
Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as:
$$
w_j(...
- 10.5k
7
votes
1
answer
364
views
Differentials in Weil model for equivariant cohomology
Why should we define the differential in Weil model as follows? I could understand $\sum_{j,k} c_{jk}^i \theta_j \wedge \theta_k$ plays a role in the formula because it is the dual of the structure ...
- 677
2
votes
0
answers
132
views
Do we have estimate like $\int_\gamma \alpha \le |\alpha| \cdot |\gamma|$? [closed]
Let $(X,g)$ be a compact smooth Riemannian manifold. It is known that $H^1(X, \mathbb R)\cong \mathrm{Hom} (\pi_1(X), \mathbb R)$, namely there is a natural pairing
$$
H^1(X) \times \pi_1(X) \to \...
- 2,789
8
votes
1
answer
313
views
Moishezon manifold vs proper complex variety
Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is ...
user132250
3
votes
0
answers
121
views
Topology of abstract varieties over $\mathbb{C}$
What are the known restrictions on the topology of complex manifolds corresponding to analytifications of smooth proper algebraic varieties over $\mathbb{C}$? I think
they have to have non-zero $b_2$ ...
user132250
6
votes
1
answer
174
views
Moishezon manifold with vanishing $b_2$
Does there exist a closed Moishezon manifold with zero second Betti number?
- 125
3
votes
2
answers
782
views
Relation between optimal transport cost and difference between topological invariants?
I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of ...
7
votes
0
answers
226
views
The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons
In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical
group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with ...
- 10.5k
29
votes
1
answer
2k
views
Does the Gauss-Bonnet theorem apply to non-orientable surfaces?
I hesitated for a long time to ask such an elementary-seeming question on Math Overflow, but when I asked and bountied it on Math SE, I found that a few experts seem to disagree on the answer, and I ...
- 1,311
5
votes
1
answer
503
views
Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure
The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...
- 10.5k
3
votes
1
answer
505
views
non-existence of global coordinates
Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad....
- 277
6
votes
1
answer
384
views
Deforming a section to a section without zeros
Let $M$ be an oriented manifold of dimension $n$. Suppose furthermore that $E$ is an oriented vector bundle of rank $n-1$ over $M$. Let $s$ be a section of $E$ transversal to the zero section in $E$. ...
- 2,830
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
16
votes
2
answers
605
views
What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?
The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...
- 63.9k
10
votes
2
answers
1k
views
Is $\Bbb S^2 \times \Bbb S^4$ symplectic?
I'm playing around with products $M = \Bbb S^{n_1} \times \Bbb S^{n_2}$, and a quick computation using the Künneth formula tells us that if $(n_1,n_2)$ is not $(1,1)$ or $(2,4)$, $M$ is not symplectic ...
- 1,163
6
votes
1
answer
679
views
Generalized projective spaces, spheres, and exotic spheres [closed]
I like to explore and ask for proper references for the relations between generalized projective spaces, spheres, and exotic spheres:
The real projective space
$\mathbb{RP}^1 \simeq S^1,$
is ...
- 10.5k
10
votes
3
answers
757
views
Spin-H structures
Let us define a Spin-H structure as a reduction of a SO(n)-bundle by the group: $$Spin^H (n)=Spin(n) \times SU(2)/\{ 1,-1\}$$ The Spin-H structures are analogous to the well-known Spin-C structures ...
- 187
7
votes
0
answers
393
views
$U(1)$ v.s. $SU(N)$ v.s. $SO(N)$ instantons
I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of:
Chern class (1st, 2nd), and
...
- 2,147
5
votes
0
answers
543
views
a question on Hodge and Atiyah's paper "integrals of the second kind on an algebraic variety"
I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71:
$$H_{2n-q}(S)...
- 1,121
7
votes
1
answer
301
views
When does $\pi:M\to M/G$ have homotopy lifting property?
Let M be an n dimensional topological manifold, may be non-compact. Suppose there is an action of a group G on M, the orbits are closed, but may not be bounded. Consider the projection $\pi: M \to M/G$...
7
votes
1
answer
223
views
Five-dimensional manifolds fibering over a fixed hyperbolic surface
I am aware of the classical work by Smale and Barden computing the diffeomorphism type of smooth simply connected 5-manifolds in D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965),...
- 2,630
10
votes
0
answers
192
views
k-th Pontryagin class of $\Lambda^{2k}_{\pm}$ on an oriented $4k$-manifold
If $M^{4k}$ is an oriented Riemannian $4k$-manifold, then the star-operator splits the bundle $\Lambda^{2k}$ into $\pm 1$-eigenspace bundles denoted $\Lambda^{2k}_{\pm}$.
I'm curious if anyone has ...
- 787
7
votes
2
answers
1k
views
Is there a sensible notion of a winding number of a closed curve in $\mathbb{R}^n$, $n\geq 3$, with respect to a point not lying on it?
I have been browsing "Topological Degree Theory and Applications" by Cho, Chen and O'Regan as well as "Mapping Degree Theory" by Outerelo and Ruiz, but I have not been able to quite answer myself the ...
- 7,127
3
votes
0
answers
98
views
Euler characteristic of an exhaustion of compacts of a surface
Let $X$ be an open (connected) Riemann surface of finite Euler characteristic. And $K_1 \subset \cdots K_n \subset$ be an sequence of closures of bounded open subsets with smooth boundary of $X.$
...
- 750
5
votes
1
answer
266
views
$S^1$-quotient of the space of unbased contractible loops of a finite dimensional $K(\pi,1)$
Let $X$ be a finite dimensional $K(\pi,1)$ manifold. Recall that the space of unbased null-homotopic loops $L_0(X)$ is the space of contractible continuous maps $S^{1}\to X$. There is an obvious $S^1$-...
- 14.3k
4
votes
1
answer
314
views
Understanding spinnable vector bundles with classifying spaces
Let $E \to X$ be an oriented vector bundle over a CW complex $X$. Suppose that the second Stiefel-Whitney class $w_2(E)=0$, then $E$ admits a spin structure. In terms of homotopy theory this means ...
- 225