6
votes
1answer
362 views

Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?

The group $\mathbb{Z}_2$ acts on $S^7\times S^7$ by switching the coordinates with fixed point set $\Delta(S^7\times S^7)\cong S^7$. I want to know whether there are some other $\mathbb{Z}_2$ actions ...
13
votes
1answer
299 views

Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...
11
votes
3answers
1k views

(Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...
4
votes
0answers
130 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
vote
0answers
75 views

Singular leaf of Strebel differential

Let $R$ be a Riemann surface.Let $\gamma$ be a loop which is non-trivial in $H_{1}(R,\mathbb{Z})$. By the Jenkins–Strebel Theorem we know the following: there exists a holomorphic quadratic ...
5
votes
2answers
256 views

Affine structures

I would like to study manifolds endowed with a linear connexion $\nabla$ which is torsion free and locally flat i.e. its curvature is $0$ (such a connexion is called flat if in addition, its holonomy ...
11
votes
0answers
336 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 ...
1
vote
1answer
173 views

Universal bundles: construction of the map associated to a group homomorphism

For a Lie group $G$ let $EG \to BG$ denote the universal bundle. A Lie group homomorphism $\rho: G \to H$ determines a map $B \rho: BG \to BH$ as the classifying map for the principal $H$-bundle $EG ...
15
votes
1answer
278 views

Cohomology class of the group extension from a principal bundle

Let $M$ be a closed connected manifold and fix a basepoint $q \in M$ and a Riemannian metric on $M$. Let $F(M)$ denote the orthonormal frame bundle of $M$. This is a principal $O(n)$-bundle over $M$ ...
11
votes
3answers
753 views

Which submanifolds are zero sets of $\mathbb{R}^n$-valued maps?

If $M$ is a smooth, compact, orientable manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal ...
7
votes
0answers
229 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. ...
0
votes
0answers
283 views

A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version. Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...
2
votes
0answers
320 views

The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)

(Migrated from MSE) While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
6
votes
0answers
141 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 ...
7
votes
1answer
232 views

How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?

This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map $$ \mu: G\times ...
3
votes
1answer
283 views

Does the cohomology after Dehn surgery depend only on the original 3-manifold or also how the knot is situated?

For $f:S^1\to M$ a knot in a 3-manifold, we can construct a 3-manifold $N$ by a $0/1$-type Dehn surgery along $f$: First remove from $M$ a solid torus which is a tubular neighbourhood of the knot ...
6
votes
2answers
391 views

Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...
3
votes
2answers
174 views

Nielsen-Thurston classification of homeomorphisms for open surfaces?

In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...
0
votes
0answers
158 views

Symplectic submanifolds in $\mathbb{R}^4$

Which symplectic submanifolds can be realized in $\mathbb{R}^4$ with standard ($\text{d}\,\boldsymbol{p} \wedge \text{d}\,\boldsymbol{q}$) symplectic structure? It's easy to show that such ...
1
vote
1answer
221 views

Possible homotopy-theoretical approach to Gauss-Bonnet

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
4
votes
2answers
214 views

Relative Serre spectral sequences?

I consider the following two situations: Let $B$ be a simply connected space, and $F\to E\to B$, $F'\to E'\to B$ two fibrations with a map $f:E'\to E$ sending fibers to fibers and inducing the ...
15
votes
1answer
409 views

Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
1
vote
2answers
246 views

The Schottky group and the fundamental group of a compact Riemann surface

I am quoting the following description from a paper, "...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...
9
votes
1answer
396 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
17
votes
2answers
437 views

A simple proof that parallelizable oriented closed manifolds are oriented boundaries?

So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
1
vote
0answers
84 views

Euler class and self-intersection number of a surface in a 4-manifold [duplicate]

In the first two paragraphs of Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings, it is claimed that For a compact oriented surface $X$ in a 4-dimensional oriented ...
2
votes
1answer
183 views

Chern class of Hopf fibration over elliptic curve

Let $N = \{ (z_0,z_1,z_2) \in S^5 \mid z_0^3+z_1^3+z_2^3 = 0 \}$, where we consider $S^5\subset\mathbb{C}^3$. The circle $U(1)$ acts on $N$ by $$e^{i\theta} \cdot (z_0,z_1,z_2) = ...
1
vote
0answers
77 views

PL or projective PL map on the links of a PL manifold

Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
1
vote
1answer
199 views

Normal tubular neighborhood theorem for semi(or pseudo)-riemannian manifolds

Suppose you have a manifold $M$ and a closed sub-manifold $A$, and let $g$ be a semi-riemannian metric,ie, $g_x$ defines a quadratic form on $T_xM$ such that $g_x(v,v)\ge0$, but $g_x(v,v)=0$ not ...
0
votes
0answers
142 views

Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...
6
votes
1answer
353 views

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 ...
2
votes
1answer
231 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
0
votes
1answer
173 views

Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$

I'll first state the question as concisely as I can and then provide some motivation. Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...
10
votes
3answers
499 views

Representing de Rham cohomology by smooth maps

It is well-known that in the homotopy category of, say, CW-complexes the singular cohomology functor is represented by the Eilenberg-Maclane spaces: $H^n(M,\mathbb{Z})=[M,K(\mathbb{Z},n)]$. My ...
2
votes
1answer
134 views

Construction of a classifying map from a connection 1-form

From a connection 1-form on $M$, I can construct a parallel transport from which in turn I can construct a classifying map $M \to BG$. Is there a construction of such a classifying map directly from a ...
1
vote
2answers
501 views

Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
1
vote
2answers
149 views

Segre class of smooth vector bundles over smooth manifolds?

Before you read the following question, please assume I have no knowledge in algebraic geometry. Is it possible to define Segre class of a smooth complex vector bundle over a smooth manifold by using ...
15
votes
2answers
846 views

Converse to Stokes' Theorem

Does satisfying Stokes' Theorem imply that a form is linear? Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : ...
1
vote
0answers
124 views

About the Lie algebra of polyvector fields

I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...
2
votes
2answers
305 views

filtration in K-theory and ordinary cohomology

I am going to ask a question, which could be a stupid one. I am reading a paper "an index theorem in differential K-theory". The first paragraph of section 8.28 recalls a filtration of K-theory ...
4
votes
2answers
221 views

good reference on brieskorn manifold

I am trying to learn something on the Brieskorn manifold (interested in the topological property) Can the Mathoverflow Experts give me some good refencece (in English)? By the way,is there an ...
6
votes
1answer
247 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. ...
3
votes
0answers
133 views

Rigidity of secondary characteristic classes

For a representation $\rho:\pi_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes $$\hat{c}_k(E_\rho)\in H^{2k-1}(M,R/Z)$$ ...
3
votes
1answer
305 views

Spin structures and divisibility of cohomology classes

Let me begin with some motivation. In calculating the Chern-Simons invariant of a $U(1)$ connection $A$ on a 3-manifold $M$, we can proceed by picking a bounding 4-manifold $X$ with $\partial X = M$ ...
15
votes
1answer
638 views

Combinatorial spin structures

I would like to know how to define spin structures combinatorially, for an oriented smooth manifold equipped with a triangulation. In the case of a 2d manifold, spin structures correspond to ...
3
votes
2answers
458 views

Is there a “by hand” proof on the symmetry of the Atiyah class of $TX$?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
-1
votes
2answers
557 views

Regarding understanding differential geometry [closed]

I am essentially looking for a book that would hold my hand through basic concepts to more complicated ones. I am coming from physics. I am looking to make some connections with Classical mechanics ...
14
votes
1answer
622 views

A concrete realization of the nontrivial 2-sphere bundle over the 5-sphere?

Since $\pi_4 (PU(2)) = \pi_4 (SO(3)) = {\mathbb Z}_2$, the two-element group, we know that half of the two-sphere bundles over the 5-sphere $S^5$ are trivial and the other half are non-trivial and ...
8
votes
3answers
271 views

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 ...
2
votes
1answer
163 views

winding number for outer-pointing normal

While trying to characterize the complexity of a closed differentiable curve (for a path planning application), I've been using a notion which is similar in spirit to the winding number of a curve. ...