0
votes
0answers
39 views

Can Vanishing Cycles be Described as Fibers over Critical Points in a Lefschetz Fibrations?

I'm trying to see if it makes sense to see vanishing cycles in a Lefschetz fibration as the fibers over critical points. A Lefschetz fibration $f: M^4 \rightarrow X$ , where $M^4$ is a smooth ...
0
votes
0answers
11 views

Orthogonal Array Planes

In an OA-plane (Orthogonal Array-Plane) how we define the touching of two lines? If we consider OA of strength of 3 with unit index , that is the plane OA-plane of rank 3 (we call this Laguerre ...
1
vote
1answer
27 views

Is there a name for the result that homotopy groups are insensitive to removing submanifolds of sufficiently large codimension?

First some motivation. Consider $\mathbb{R}^n-\{0\}$. This is simply connected iff $n > 2$, since it deformation retracts to $S^{n-1}$. If instead we consider $\mathbb{R}^n - L$ where $L$ is a ...
2
votes
3answers
99 views

Is it possible a trivial fiber bundle with nonzero holonomy?

Let $P\rightarrow M$ be a principal bundle with structure group $G$. Suppose that the bundle is trivial $M\times G$; is it possible to have a nonzero holonomy along some closed trajectory on $M$ for ...
7
votes
1answer
130 views

Uniqueness of curve of minimal length in a closed $X\subset \mathbb R^2$

Suppose $X$ is a simply connected closed subset of $\mathbb R^2$. Let $a,b$ belong to $X$. Is it true that there is at most one curve inside $X$ from $a$ to $b$ such that the length of the curve is ...
10
votes
1answer
102 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
6
votes
1answer
69 views

De Rham cohomology of $T^*\mathbb{CP}^n$

I am a bit rusty on my de Rham cohomology, and I'm hoping that someone here could help me. I want to find the cohomology of $T^*\mathbb{CP}^n$ (seen as a real manifold). Now, this should be equal to ...
6
votes
1answer
56 views

Relation between different definitions of degree in complex geometry

Consider a holomoprhic map from a Riemann surface $$ f: \Sigma_g \to \mathbb{CP}^n. $$ This is given by some homogeneous polynomials in some variables. How can we show that the homogeneous degree $d$ ...
2
votes
1answer
51 views

vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
1
vote
1answer
55 views

Projection map between the Stiefel manifold and the Grassmanian

I am trying to show that the projection map $\pi: V_{k}(\mathbb{R}^{n+k}) \rightarrow \mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ is a fiber bundle with fiber $O(k)$, the group of orthogonal $k \times ...
6
votes
2answers
124 views

Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such ...
0
votes
3answers
85 views

Top Cohomology of $\mathbb{P}^2$ via Sphere

I am trying to use the cohomology of the sphere to calculate $H^2(\mathbb{P}^2)$. My professor just mentioned there's an argument using the projection $\pi: \mathbb{S}^2 \to \mathbb{P}^2$ and the ...
1
vote
1answer
68 views

Cohomology group of a torus with g holes

I have to compute the cohomology groups of a torus with g holes (the Riemann surface of genus g). first I have computed the cohomology of a Torus with 3 holes in the following way: I pick a covering ...
1
vote
0answers
42 views

Consequences of tubular neighbourhood theorem

Consider an oriented manifold without boundary S embedded in an oriented manifold M. The tubular neighbourhood theorem says that there is a neighbourhood T of S in M which is diffeomorphic to ...
3
votes
0answers
42 views

differential forms on covering spaces

I seem to be under the impression that if $p:A \to B$ is a regular covering (of smooth manifolds) with $\alpha\in \Omega^k(A)$; there exists $b\in \Omega^k(B)$ such that $\alpha= p^*\beta$ if and only ...
4
votes
0answers
116 views

Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...
2
votes
1answer
34 views

integration of differential forms on covering space

Let $M_1,M_2$ be $n$-dimensional oriented manifolds. Let $f: M_1\longrightarrow M_2$ be an orientation-preserving diffeomorphism. Then for any $\omega\in \Omega^n_c(M_2)$ we have(page 85 of {Madsen: ...
4
votes
0answers
95 views

is the image of a polynomial map contractible?

I asked this in MSE http://math.stackexchange.com/questions/643348/is-the-image-of-a-polynomial-map-contractible and got no response. Either it's a silly question or I posted it under the wrong ...
0
votes
1answer
61 views

Intuition on Whitney–Graustein theorem

According to the Whitney–Graustein theorem, two regular curves are regularly homotopic if and only if their winding numbers are the same. Suppose I have a circular curve but with an extra loop so ...
5
votes
1answer
82 views

How do Chern classes behave under connected sums?

I often see people talking about Chern classes of manifolds like $\mathbb{CP}^2 \# \mathbb{CP}^2$, or $\mathbb{CP}^2\#\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$, where $\#$ denotes connected sum and the ...
5
votes
1answer
85 views

Lower bound for the size of an atlas

This question came up in a graduate-level class on differential topology I'm currently taking; the instructor couldn't come up with an answer off the top of her head and while I'm very new to the ...
2
votes
2answers
80 views

Are these two 2-manifolds homeomorphic?

I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...
5
votes
1answer
47 views

When can a connection be lifted?

Let $P \rightarrow X$ be a principal $G$-bundle, and $P' \rightarrow X'$ be a principal $G'$-bundle. Let $(f',f'')$ be a morphism from $P'$ to $P$, i.e., a pair of maps $f': P' \rightarrow P$ and ...
2
votes
0answers
27 views

Difference between diffeomorphisms fixing a point or a whole neighborhood.

Let $S_g$ be a closed orientable surface of genus $g$ and $S_{g}^1$ a closed orientable surface with one boundary component. Let $p$ be in $S_g$ and let's note $\mathrm{Diff}_+(S_g,p)$ the set of ...
6
votes
2answers
335 views

how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without ...
2
votes
1answer
32 views

isometric imbedding of the projective plane

How to isometrically imbed the projective plane (identifying antipodal points of the unit sphere) in $\mathbb{R^5}$? My textbook indicates that there is an isometric imbedding of the projective ...
0
votes
2answers
57 views

Isometry of surfaces in $\mathbb{R}^3$

Let $F$ be an isometry of the Euclidean space $\mathbb{R}^3$. Hence $F$ is orthogonoal transform followed by translation by a constant vector. Let M be a surface of $\mathbb{R}^3$ that is connected, ...
3
votes
2answers
48 views

When is a regular map a covering map

Let M, N be two manifolds of the same dimension A map from M to N is regular provided its tangent map is one to one. A map from M to N is a covering map provided each point in N has a neighborhood ...
1
vote
0answers
73 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
2
votes
0answers
59 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 ...
2
votes
1answer
52 views

compact surface with two non-intersecting geodesics

I need to find an example of a compact geometric surface M such that Gaussian curvature $K>=0$ M is diffeomorphic to a sphere M has two simply closed geodesics (smoothly closed loops) that ...
6
votes
0answers
284 views

Trying to Understand Lefschetz Pencils

I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) better, tho I would appreciate insights on condition i), and in general. A Lefschetz pencil on a $4$-manifold $X$ is a ...
1
vote
0answers
40 views

homeomorphism still isotopic to the identity after the deletion of points.

I have a surface $M$ (without boundary) and a homeomorphism $f:M \rightarrow M$, which is isotopic to the identity on $M$. If I delete two points $x$ and $f(x)$ from the surface, I get a ...
1
vote
2answers
77 views

Frame bundle of orthonormal frames orthogonal to a submanifold.

Suppose we have a smooth manifold $M$ of dimension $m$ with a Riemannian metric and a connected submanifold $N$ of dimension $n$ in $M$ with $n<m-1$. Let $n\le k<m-1$ and consider the bundle ...
2
votes
1answer
35 views

Definition of a one-connected manifold?

Perhaps the question is self-explanatory. The context is Kleiner's Inv. Math. paper An isoperimetric comparison theorem, where the statement of the main theorem begins with "Let $M$ be a complete ...
1
vote
0answers
40 views

Relative de Rahm cohomology computation for two disjoint circles embedded in R^2

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
1
vote
2answers
86 views

Definition of a $n$ - form on a manifold

I am confused about the definition of a differential form on a manifold. The definition I have comes from Bott and Tu and is as follows: A differential form, $\omega$, on a manifold $M$ is a ...
1
vote
2answers
99 views

Calculate the Euler Characteristic of M

Let $M$ be the following subsets of $\mathbb R^4$:$$M= \{(x,y,z,w), 2x^2+2=z^2+w^2, 3x^2+y^2=z^2+w^2 \}$$ we know $M$ is a submanifold of $\mathbb R^4$,what is the Euler Characteristic of M?
0
votes
0answers
90 views

Vector field and triangulation

I'm writing a paper on Poincaré-Hopf theorem about vector field indices and Euler Characteristic over topological compact surfaces and I got struck finding details on the last part of the proof. Let ...
1
vote
0answers
44 views

finite covering space of non-orientable surfaces

Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers. A ...
3
votes
1answer
175 views

Intuition of Chern-Weil theory

Let $ P \rightarrow M$ be a $G$-principal bundle. The lie algebra of $G$ is $\frak{g}$ and $P$ has connection form $\omega \in H^1(P,\frak{g})$ and curvature form $\Omega \in H^2(P,\frak{g})$. We ...
1
vote
2answers
96 views

Is there any standard N-sphere that has non-trivial first Pontryagin class?

I am wondering if there is a standard $N$-sphere that has non-trivial first Pontryagin class on its tangent bundle $TS^n$and frame bundle $FS^n$? I know that only $S^4$ has non-trivial $H^4(S^n, R)$ ...
2
votes
1answer
89 views

Lift of a diffeomorphism of the Torus

I'm trying to prove the following formula. Suppose to have $p:\mathbb{R}^{d}\rightarrow\mathbb{T}^{d}$ the canonical projection of the real d- dimensional space in to the d-dimensional torus, and ...
13
votes
1answer
219 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
9
votes
2answers
124 views

If a smooth manifold X is covered by an odd sphere, then X is orientable.

In solving some old qualifying exam questions, I've been thoroughly stumped. If a smooth manifold $X$ is covered by an odd dimensional sphere, then $X$ is orientable. I see this question has ...
3
votes
0answers
52 views

Does the boundary of a handle decomposition obtain a handle decomposition?

Let $M$ be the $4$-manifold $D^4\cup2\text{-handle}\cup\ldots\cup2\text{-handle}$, where the attachment of the handles is specfied by an oriented framed link $L=L_1\cup\ldots\cup L_n\subseteq S^3$. By ...
3
votes
1answer
42 views

The smooth deformation

$M$ is a connected smooth manifold and $p$ is a fixed point on $M$. For a null-homotopic smooth loop $\gamma$ at $p$, can we find a smooth deformation, that is, a smooth function $f :[0,1] \times ...
0
votes
1answer
85 views

On an open set $U ⊂ \mathbb{R}^n$ show that the exterior derivative d is the only operator $d : Ω^p(U) → Ω^{p+1}(U)$ satisfying

On an open set $U ⊂ \mathbb{R}^n$ show that the exterior derivative $d$ is the only operator $d : Ω^p(U) → Ω^{p+1}(U)$ satisfying: $d(ω + η) = dω + dη$; $ω ∈ Ω^p(U), η ∈ Ω^q(U) ⇒ d(ω ∧ η) = dω ∧ η + ...
4
votes
1answer
152 views

Turning higher spheres inside out

I know that one can turn a sphere $S^2$ in $\mathbb{R}^3$ inside-out having at each time an immersion of $S^2$ into $\mathbb{R}^3$. It is called Smale paradox. There is beautiful animation about that. ...
7
votes
0answers
138 views

De Rham Cohomology of $M \times \mathbb{S}^1$

Let $M$ be a closed (compact, without boundary) $m$-dimensional manifold. I want to prove that $H^{k+1}(M \times \mathbb{S}^1) = H^k(M) \oplus H^{k+1}(M)$. ($H^k$ is the $k$-th De Rham cohomology ...