# Tagged Questions

**4**

votes

**1**answer

127 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

**2**answers

134 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

**0**answers

134 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

**1**answer

195 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

**2**answers

185 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

**1**answer

387 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

**2**answers

221 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

**1**answer

348 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 ...

**10**

votes

**0**answers

150 views

### A simple proof that parallelizable oriented closed manifolds are oriented boundary?

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

**0**answers

80 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

**1**answer

154 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

**0**answers

74 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

**1**answer

133 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

**0**answers

137 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

**1**answer

243 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

**1**answer

220 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

**0**answers

74 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

**3**answers

482 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

**1**answer

130 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

**2**answers

476 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?

**0**

votes

**2**answers

140 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

**2**answers

760 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

**0**answers

119 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

**2**answers

295 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

**2**answers

208 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

**1**answer

216 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.
...

**4**

votes

**0**answers

118 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

**1**answer

285 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 $\partial X = M$ ...

**16**

votes

**1**answer

568 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

**2**answers

437 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 ...

**-2**

votes

**2**answers

500 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

**1**answer

586 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 ...

**7**

votes

**3**answers

225 views

### A version of Lusternikâ€“Schnirelmann category for good open covers

Recall that the Lusternikâ€“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

**1**answer

151 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. ...

**5**

votes

**2**answers

259 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 KÃ¼nneth theorem that allows one to compute the cohomology of the fibred product?

**7**

votes

**2**answers

280 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 ...

**12**

votes

**2**answers

785 views

### Converse of Poincaré-Hopf theorem

Let $M$ be a connected, compact, oriented manifold of dimension $n<7$. If any two maps $M \to M$ having equal degrees are homotopic, must $M$ be diffeomorphic to the $n$-sphere?

**4**

votes

**2**answers

252 views

### Cohomology of submanifold complements

Let $X$ be a finite-dimensional complex manifold (possibly non-compact). Let $\mathcal{H}$ be a union of codimension-$1$ submanifolds such that the local picture is that of intersecting hyperplanes. I ...

**7**

votes

**0**answers

169 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 ...

**3**

votes

**1**answer

223 views

### Self-intersection of immersed surfaces and connected sum

Let me first say that my background is theoretical physics so I find it hard to look at some of the mathematical literature.
The kind of problem I am interested in is the following one. Consider a ...

**18**

votes

**1**answer

783 views

### Example of 4-manifold with $\pi_1=\mathbb Q$

This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.

**1**

vote

**0**answers

105 views

### Number of generators of the fundamental group of a Riemannian manifold with Ricci curvature bounded below

Is there a constant $C(n,D)$ such that for any closed Riemannian manifold $M$ with $Ric \ge -(n - 1)$ and $\mathrm{diam} \le D$, the fundamental group $\pi_1(M)$ is generated by at most $C(n,D)$ ...

**3**

votes

**1**answer

228 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 $1$?

**1**

vote

**1**answer

208 views

### equivariant euler class

Let $\pi:E\longrightarrow B$ be a $G-$vector bundle and $s:B\longrightarrow E$ be an equivariant smooth section such that $s^{-1}(0)$ is compact, where $G$ is a compact Lie group. From classical ...

**2**

votes

**1**answer

134 views

### simple explaination of simplicial volume=4g-4 when genus $\ge 1$

In Gromov's famous book ,it says "simplical volume of every oriented surface of genus $ \ge 1$ satisfies${\left\| {\left[ S \right]} \right\|_\Delta } = 4g - 4 = - 2\chi \left( S \right) = - 2({k_0} ...

**0**

votes

**2**answers

219 views

### What does a singular simplex with real coefficient mean [closed]

For an $n$-dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$-norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which represent the fundamental ...

**1**

vote

**2**answers

396 views

### fundamental class is the sum of simplices of triangulation of the manifold?

M is an n-dimensional closed orientable manifold. I find in a book "Intuitively,the fundamental class can be thought of as the sum of the (top-dimension) simplices of a suitable triangulation of the ...

**3**

votes

**3**answers

274 views

### Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings

If $X$ is a compact oriented surface in a 4-dimensional oriented manifold $M$, then the self-intersection number $X^2$ of $X$ is given by the integral over $X$ of the Euler class of the normal bundle. ...

**9**

votes

**0**answers

397 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 ...

**7**

votes

**1**answer

326 views

### Classification of higher dimensional manifolds

It is known that a 2-connected closed smooth 6-manifold is homeomorphic to S^{6}
or connected sum of (S^{3}xS^{3}). My question is whether we have a similar statement for (n-1)-connected closed ...