All Questions
Tagged with dg.differential-geometry at.algebraic-topology
172 questions with no upvoted or accepted answers
63
votes
0
answers
2k
views
Are there periodicity phenomena in manifold topology with odd period?
The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...
- 118k
50
votes
0
answers
12k
views
Atiyah's paper on complex structures on $S^6$
M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$.
https://arxiv.org/abs/1610.09366
It relies on the topological $K$-theory $KR$ and in ...
- 9,870
48
votes
0
answers
17k
views
What is the current understanding regarding complex structures on the 6-sphere?
In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
- 2,995
20
votes
0
answers
789
views
Is the determinant of cohomology a TQFT?
If $M$ is an oriented $d$-manifold, let $D(M)$ denote the top exterior power of $H^*(M,\mathbf{C})$. Then $D(M_1 \amalg M_2) = D(M_1) \otimes D(M_2)$. Is there a good recipe for a map $D(M) \to D(N)$...
- 4,949
18
votes
0
answers
496
views
Orientation-reversing homotopy equivalence
If there is an orientation-reversing homotopy equivalence on a closed simply-connected orientable manifold is there an orientation-reversing homeomorphism?
It is not true, for instance, that if there ...
user164740
18
votes
0
answers
503
views
Lipschitz constant of a homotopy
Let $n>0$, $\mathbb S^n$ be $n$-sphere and $1\in \mathbb S^n$ be its north pole.
A am looking for an example of compact manifold $M$ with a continuous $n$-parameter family of maps $h_x\colon M\to ...
- 45k
16
votes
0
answers
425
views
Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
- 587
15
votes
0
answers
1k
views
Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle
My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
- 648
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
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
votes
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
0
answers
1k
views
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
0
answers
650
views
Triangulation of manifolds with corners
Let's begin with some definitions:
A (smooth) manifold with corners is a Hausdroff (and second countable if you want) space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \...
- 909
11
votes
0
answers
1k
views
Equivariant and orbifold Chern classes
Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved ...
- 6,813
10
votes
0
answers
342
views
Hodge structure and rational coefficients
Suppose $X$ is a complex projective variety with a model $X_\mathbb{Q}$ defined over the rational numbers. Then there is a rational de Rham lattice $H^k_{dR}(X_\mathbb{Q}, \mathbb{Q})\subset H^k(X, \...
- 7,952
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
10
votes
0
answers
6k
views
Atiyah's paper "Non-existent complex 6-sphere"
I'm trying to understand the main idea of Atiyah's proof (https://arxiv.org/abs/1610.09366). Although there were discussions on MO year ago I couldn't find answers to my questions.
Consider the ...
- 171
10
votes
0
answers
365
views
diameter as a Morse function
Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...
- 16.6k
10
votes
0
answers
739
views
What is Quillen's contribution to index theorem?
In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...
- 9,019
10
votes
0
answers
281
views
Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$
Is there any example of a compact manifold $M$ of dimension $n>10000$
such that
$M$ admits an embedding into $\mathbb R^{n+2}$,
$M$ is hyperbolic; i.e., it admits a Riemannian metric with
...
- 28.9k
9
votes
0
answers
338
views
Is $\beta^{*}(w_{2k-2}) = 0$ for an open orientable $2k$-manifold?
This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case.
Let $M$ be a ...
- 1,726
9
votes
0
answers
570
views
In terms of sheaf cohomology, what does Bott & Tu's relative de Rham cohomology $H^\bullet(f)$ compute for $f: S \to M$ a smooth map?
Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where ...
- 6,025
9
votes
0
answers
268
views
Chern-Simons form and Rarita-Schwinger operator
The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2.
I was wondering if there exists any reference concerning the ...
- 405
9
votes
0
answers
408
views
Why does the index of the Dirac operator on a manifold with boundary live inside the Pfaffian line of the boundary Dirac operator?
I am trying to understand a statement from page 72 of What is an elliptic object? by Stolz and Teichner.
They have a spin Riemannian 2-manifold $\Sigma$, with boundary 1-manifold $Y$. The Dirac ...
- 6,829
8
votes
0
answers
232
views
+300
Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
- 491
8
votes
0
answers
251
views
(Higher) flat connections and Grothendieck construction
For any (nice) topological space there is an equivalence between covering spaces and local systems $\pi_1X \to \operatorname{Set}$. We can think of it as a Grothendieck construction. If we take ...
- 834
8
votes
0
answers
332
views
Differential version of $G\mapsto H^3(G,\mathbb Z)$?
Let $\mathit{cLieGrp}^{\mathrm{inj}}$ be the category of compact connected Lie groups, and injective continuous group homomorphisms.
Is there a reasonable functor (some kind of degree $3$ differential ...
- 43.2k
8
votes
0
answers
281
views
Combinatorial spin$^{\mathbf{C}}$ structures
Below is a brief introduction to spin$^{\mathbf{C}}$ structure that I took from Wikipedia. For more information, one should refer to https://en.wikipedia.org/wiki/Spin_structure#SpinC_structures.
A ...
- 875
8
votes
0
answers
289
views
Geometric meaning of Aomoto complex
Generally, if we have any space $X$, then multiplication by any odd class $\eta$ makes $H^*(X)$ a complex (usually called Aomoto complex) $A_{\eta}$ because cup product is graded commutative. It is ...
- 4,600
8
votes
0
answers
234
views
Explicit diffeomorphism between an infinite dimensional sphere its product with itself
Let $S$ be an infinite dimensional sphere in a Hillbert space.
As $S$ is homotopic to the product $S \times S$, then $S$ is diffeomorphic to $S \times S$ (for Hilbert manifolds, a homotopy ...
- 606
8
votes
0
answers
352
views
Two proofs of the Cheeger-Müller theorem
In the late 1970's, Cheeger and Müller independently proved the equality of analytic torsion and Reidemeister torsion for orthogonal representations, which had been conjectured by Ray-Singer. Their ...
- 266
8
votes
0
answers
463
views
On the cohomology of Kontsevich graph complex
Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...
- 1,609
8
votes
0
answers
285
views
Systoles of hyperbolic (Riemann) surfaces of large genus
Let $m$ be a Riemannian metric on $S_g$ the surface of genus $g$, and $sys(m)$ be the length of the shortest non contractible cycle with respect to $m$.
The systolic inequality claims that for any ...
- 1,303
8
votes
1
answer
1k
views
semi flat connections
Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.
For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\...
7
votes
0
answers
194
views
Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
- 44.7k
7
votes
0
answers
350
views
Do smooth maps with nowhere-maximal rank have small image?
I’m trying to better understand the concept of “maps with small image” as used by Lipyanskiy in his construction of “geometric homology” in https://arxiv.org/abs/1409.1121. Lipyanskiy utilizes ...
- 5,489
7
votes
0
answers
253
views
Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$
The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by
$$
\eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
- 91
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
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
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
7
votes
0
answers
655
views
de Rham cohomology with twisted coefficients and group cohomology
I'm trying to follow Goldman's paper 'The symplectic nature of fundamental groups of surfaces'. I'm having trouble understanding the following isomorphism: let's take a principal $G$-bundle $P \to M$ ...
- 279
7
votes
0
answers
359
views
Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices
Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $...
- 186
7
votes
0
answers
484
views
manifold branched covering space for orbifolds
An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...
7
votes
0
answers
199
views
Central Extension of Continuous Loop Group
For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the ...
- 1,040
6
votes
0
answers
129
views
Are there isospectrally equivalent exotic spheres?
Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?
I would be happy ...
6
votes
0
answers
297
views
Regarding homology of fiber bundle
Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is ...
- 585
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
6
votes
0
answers
231
views
Does $\#_n S^2×S^1$ really admit a map of non-zero degree from $B×S^1$?
In Proposition 4 on page 6 of this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ ...
- 61
6
votes
0
answers
621
views
Gompf's invariant of $2$-plane fields
I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...
- 369