All Questions
Tagged with dg.differential-geometry at.algebraic-topology
639 questions
9
votes
1
answer
554
views
Restrictions of perfect Morse functions to submanifolds
A Morse function $f: M \rightarrow \mathbb R$ on a connected closed manifold $M$ is called $\mathit{perfect}$ with respect to the field $\mathbb F$ if all of the Morse inequalities are equalities, i....
9
votes
2
answers
405
views
Differential refinement of homology
Differential cohomology is a refinement of ordinary cohomology by differential data. It's construction comes down to the observation that $H^2(M, \mathbb{Z})$ is isomorphic to the space of isomorphism ...
- 5,824
9
votes
2
answers
892
views
What is this analogy between manifolds and bundles (or schemes and locally free sheaves)?
There's a kind of analogy between the way manifolds work and the way bundles work. Let me try to give some examples of the analogy (although there may be better ones). I'll stick to smooth manifolds ...
- 63.9k
9
votes
1
answer
517
views
What does positivity of the first Pontryagin number of a vector bundle tell us?
Some context:
In the theory of compact, oriented Riemannian Einstein 4-manifolds, there is a a fundamental topological constraint that is implied by the Einstein equations. To wit, if $\chi$ and $\...
- 787
9
votes
1
answer
561
views
"Mathai-Quillen-type" form on $M\times M$?
Let $(M,g)$ be a compact, oriented, $(2n)$-dimensional Riemannian manifold. I'm wondering whether there is a "canonical" construction of a $(2n)$-form $\eta_g$ on $M\times M$, such that
$\eta_g$ is ...
- 3,212
9
votes
1
answer
656
views
Chern-Simons forms, characteristic numbers, and boundary terms?
For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...
- 6,025
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
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0
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570
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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
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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
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0
answers
408
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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
3
answers
3k
views
Pullback map in homology
I'm interested in a concrete description of the "wrong way maps" in homology/cohomology.
$\textbf{Question 1:}$ Let $X, Y$ be compact smooth manifolds of dimensions $n, m$ respectively, and $\phi: X \...
- 113
8
votes
2
answers
1k
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an example of a Morse-Bott function
Hi,
I want to find an example of a Morse-Bott function such that for at least one of the critical submanifolds, the orientation sheaf O is nontrivial.
- 81
8
votes
3
answers
976
views
Examples and properties of spaces with only trivial vector bundles
Let $B$ be a paracompact space with the property that any (topological) vector bundle $E \to B$ is trivial. What are some non-trivial examples of such spaces, and are there any interesting properties ...
- 1,763
8
votes
2
answers
2k
views
Why is the mapping class group of hyperbolic manifolds finite?
Hi! I'm trying to understand why a hyperbolic n-manifold has finite mapping class group if $n \geq 3 $. In books I'm reading it's said it's a consequence of Mostow's rigidity theorem:
"If M and N are ...
- 425
8
votes
3
answers
914
views
Spectral sequences in algebraic topology [duplicate]
What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a ...
8
votes
1
answer
689
views
Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle?
Let $S \to X$ be an $S^3$-fiber bundle over a smooth manifold $X$. If $S$ is an oriented manifold does this fiber bundle admit the structure of an $SU(2)$-principal bundle?
There is a similar theorem ...
- 1,716
8
votes
2
answers
2k
views
Rank 2 vector bundles over $\mathbb CP^2$
Is there any classification of the rank 2 complex vector bundles over $\mathbb CP^2$ up to diffeomorphism?
- 1,915
8
votes
3
answers
3k
views
Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?
Hello,
I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the ...
- 83
8
votes
2
answers
589
views
Easy proof of topological property of Zoll manifolds
It is known that the cohomology ring of a Zoll manifold---a riemannian manifold all of whose geodesics are periodic with the same minimal period---must be the same as the cohomology ring of a compact ...
- 13.5k
8
votes
1
answer
2k
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What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?
In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...
- 16.6k
8
votes
2
answers
1k
views
Künneth theorem for fibred products
Given the fibred product of two manifolds over a base space $X\times_Y Z$ , is there an analogue of Künneth theorem that allows one to compute the cohomology of the fibred product?
- 154
8
votes
2
answers
827
views
Any text book or lecture notes regarding the algebraic part of geometry?
I know there are text books of Algebraic topology. There are books of Differential geometry. But when I read papers, for example lots of papers talking about fundamental groups or higher homotopy ...
- 2,623
8
votes
1
answer
784
views
Cobordism invariants: topological v.s. geometric
Some cobordism invariants are not cohomology classes. Such as the $\mathbb{Z}_{16}$-valued eta invariant $\eta$ of $\Omega_4^{Pin^+}$, the $\mathbb{Z}_8$-valued Arf-Brown-Kervaire invariant ABK of $\...
- 1,329
8
votes
2
answers
896
views
Can you do geometry with persistent homology?
Setup
In practice, persistent homology of data $X$ is often used to infer the homology of the underlying (Riemannian) manifold $M$ that the data is sampled from.
However most filtrations (Vietoris, ...
- 159
8
votes
1
answer
379
views
Existence of certain "nondegenerate" function and manifold topology
Let $M$ be a smooth manifold without boundary, not necessarily compact.
Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local ...
- 783
8
votes
1
answer
530
views
Whitney's approximation theorem for Lipschitz manifolds
In the smooth setting, Whitney's approximation theorem says the following: If $M,N$ are smooth manifolds and $f,g:M\to N$ are smooth functions that are continuously homotopic (ie there is a continuous ...
- 1,149
8
votes
1
answer
426
views
Orbifolds are Thom-Mather stratified spaces
Where can I find a proof of (or if it is even true) that an (effective) orbifold is a Thom-Mather stratified space?
edit: after some search, I found the proof should be contained in either
GIBSON, C....
- 803
8
votes
1
answer
228
views
Isomorphisms of Pin groups
My goal is to identify the $Pin$ group
$$
1 \to Spin(n) \to Pin^{\pm}(n) \to \mathbb{Z}_2 \to 1
$$
such that $Pin^{\pm}(n)$ are isomorphisms to other more familiar groups.
My trick is that to look at ...
- 10.5k
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
8
votes
1
answer
2k
views
Curvature of a principal bundle and the exterior covariant derivative
I am sorry if this is too elementary; I had posted it on math.stack but no one answered.
Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a ...
- 2,816
8
votes
1
answer
474
views
$\pi_0${plane fields}$\to\mathbb{Z}_2$
On a 3-manifold $Y$, oriented 2-plane fields $\xi$ are oriented rank-2 subbundles of $TY$. Denote the set of such (up to homotopy) by $\Theta=\pi_0\lbrace\xi\rbrace$. What is an explicit canonical map ...
- 17.5k
8
votes
1
answer
469
views
Computing H^2(X, T_X(-\log D))
Let $X\subset \mathbb P^3$ be a smooth variety and $D \subset X$ be an effective, reduced, irreducible divisor. My question is the following.
If I know the defining equations of $X$ and $D$ then is ...
- 325
8
votes
1
answer
1k
views
Relative version of de Rham cohomology with local coefficients
Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$\mathcal{E} \xrightarrow{d^\nabla=\nabla} \Omega^1_M \otimes_{\...
- 6,025
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
8
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0
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234
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
1
answer
452
views
Non-algebraic Kähler threefolds with abelian $\pi_1$ of arbitrarily large rank
Do there exist non-algebraic Kähler threefolds with abelian $\pi_1$ of arbitrarily large rank?
user164740
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
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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
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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
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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
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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
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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
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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
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
7
votes
2
answers
629
views
Can one give an immersion of exotic sphere $S^7$ in a standard sphere $S^8$ of radius $1$?
Can one give an immersion of exotic sphere $S^7$ in a standard sphere $S^8$ of radius $1$?
- 91
7
votes
2
answers
1k
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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
7
votes
1
answer
692
views
Homotopically trivial vs isotopically trivial diffeomorphisms
Let $M$ be a manifold. Let's say $M$ is smooth, connected, oriented. We can also assume that $M$ is closed if that makes things easier.
Let $\mathit{Diff}(M)$ denote the group of diffeomorphisms of $...
- 1,347
7
votes
1
answer
456
views
Space-discriminating injective curve
Let $f\colon \mathbb R^1\to \mathbb R^3$ be a continuous and injective map. Is $\mathbb R^3\setminus f(\mathbb R^1)$ a path-connected space?
- 5,053