2
votes
0answers
23 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
3
votes
0answers
51 views

De Rahm Cohomology of Complex Grassmannian

Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ ...
2
votes
0answers
35 views

Homology of smooth loop spaces of spheres

I'm looking for a reference on the homology of $C^\infty(S^1,S^n)$. So far I've only been able to find references for spaces of maps which are $C^k$ or in a Sobolev class. I also have references which ...
4
votes
2answers
69 views

What is the calculus-theoretic formula to calculate the homotopy class/degree of a map $T^2\to S^2$?

I know by Hopf classification theorem that $[T^2;S^2]$(torus to sphere) are classified by the integral cohomology group $H^2(T^2;\mathbb{Z})\approx\mathbb{Z}$. Also I understand that in general, given ...
0
votes
0answers
42 views

Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
3
votes
1answer
77 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
1
vote
0answers
68 views

differential forms and orientation in Bott and Tu

I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it. I have gotten to page 29. I am confused about proposition ...
3
votes
0answers
36 views

Minimum regularity Of Stoke's theorem to hold in smooth manifold.

Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary ...
1
vote
1answer
64 views

Decomposition of cohomology group on $S^{n}$

If we have decomposition of cohomology group on $S^{n}$ it looks like $H^{n}(S^{n})=H^{n}(S^{n})_{+}\oplus H^{n}(S^{n})_{-}$, where $H^{n}(S^{n})_{\pm}$ cohomology of invariant or anti-invariant $n$ ...
1
vote
1answer
34 views

Winding number from complex analysis and differential geometry

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
2
votes
1answer
56 views

the relation between cohomology and homomorphism

I meet a problem, how can I understand $H^1(M,\mathbb{R})\cong Hom(\pi_1(M),\mathbb{R})$? Where $M$ is a compact manifold. Thanks in advance.
0
votes
0answers
36 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
2
votes
1answer
62 views

De Rham cohomology for $\mathbb{R^2}$

De Rham cohomology groups for $\mathbb{R^2}$. $H^{0}_{dR}(\mathbb{R}^{2})=\mathbb{R}$ since $Z^{0}(\mathbb{R}^{2})$ is the one dimensional space of locally constant functions on $\mathbb{R}^{2}$ and ...
5
votes
1answer
128 views

Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. [closed]

We have a quite some characteristic classes: Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc. I wonder whether some math experts can use simple words and basic ...
5
votes
2answers
222 views

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
5
votes
0answers
127 views

Which manifolds are zero sets of $\mathbb R^n$ valued maps

If $M$ is a smooth 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 bundle induced by $f$. ...
1
vote
1answer
133 views

Another differential topology lemma

Another lemma (1) Why can we assume $z=f(z)=0$ and that $U$ is convex? (the coordinate domains of the manifolds can be taken to be balls?) (2) Why is it enough to consider the special case of a ...
0
votes
1answer
166 views

A differential topology lemma

Consider the following lemma (1) How come he talks about degrees here, after all he doesn't assume $X$ to be oriented? (2) Why is $\bar{v}|\partial X$ homotopic to $g$? (NOTE: we consider them as ...
1
vote
2answers
83 views

Homology of manifolds with boundary

If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is ...
3
votes
0answers
36 views

The Morse complex of a manifold with boundary

For a smooth manifold with boundary $M$ and $\partial M = V_+ \cup V_-$ two disjoint sets of boundary components, one usually defines the Morse complex of $M$ using a Morse-Smale pair $(f,X)$ such ...
1
vote
1answer
62 views

Tangent bundle of P^n and Euler exact sequence

I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has $$ T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1} $$ where $L$ ...
0
votes
1answer
27 views

Alexander duality formulation + Jordan-Brouwer separation

In Davis & Kirk LNAT p.71 there is written: (1) How does this imply the Alexander duality $\tilde{H}^k(A)\cong \tilde{H}_{n-k-1}(\mathbb{S}^n\!\setminus\!A)$? (2) Is it assumed that the ...
7
votes
2answers
100 views

Chern-Weil: why do we divide by $2\pi$?

So here's a somewhat incoherent question. To define characteristic classes in the Chern–Weil way, one takes a curvature form $\Omega$ on a vector bundle $E \to M$ and an invariant polynomial ...
0
votes
0answers
17 views

spin^c structures and charged spinors

Given a spin structure and a complex line $\mathcal{L}$ we can form the tensor product of the complex spinor bundle $S$ and this line $S\otimes\mathcal{L}$. A spin^c structure attempts to construct ...
0
votes
0answers
25 views

express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
8
votes
1answer
203 views

Lie groups as manifolds

In Weinberg's Classical Solutions in Quantum Field Theory, he states Lie groups, such as $SU(2)$ or $SO(3)$, may be viewed as a manifold. My questions are, If we can interpret, e.g. $SU(2)$ as a ...
7
votes
3answers
127 views

let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
2
votes
2answers
106 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
0
votes
0answers
48 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
13 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
30 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
160 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
133 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
117 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
82 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
60 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
54 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
78 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
141 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
96 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
104 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
62 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
43 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
131 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
36 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
113 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
68 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
96 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
99 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
97 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 ...