0
votes
0answers
13 views

whitney class of fiber bundles over classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $\rho: \Sigma_k\to GL(k)$ be regular representation by permuting basis. Let $\rho': B\Sigma_k\to BGL(k)=G_k(\mathbb{R}^\infty)$ be the induced ...
0
votes
1answer
23 views

cup product of stiefel-whitney class

Let $\xi$ be a vector bundle. Let $w(\xi)$ be the total Stiefel-whitney class. Let $\bar w$ be the dual Stiefel-whitney class. In John Milnor's Characteristic class book, page 40-41 Chap.4, ...
4
votes
1answer
53 views

Manifold allowing function with two critical points is sphere

The only closed manifolds which allow a function with two (maybe degenerate) critical points are spheres. In dimension 2 it is quite easy to prove, but what is about higher dimensions?
2
votes
0answers
49 views

Degree of odd mapping of sphere

Is it possible to prove the fact that every smooth odd mapping of $S^n$(such that $f(x)=-f(-x)$ for every $x$) has odd degree using formula which connects degree and number of preimages of regular ...
1
vote
3answers
58 views

does a commutative diagram implies pull-back?

Let $\xi=(E,p,B),\xi'=(E',p',B')$ be fibre bundles. Let $f: B\to B'$, $\bar f: E\to E'$ be maps such that the diagram commutes $\require{AMScd}$ \begin{CD} E @>\displaystyle\bar f>> E'\\ ...
0
votes
1answer
26 views

Signature of a finite covering space

Suppose $\tilde{M}\rightarrow M$ is a finite (k-fold) covering of the smooth, oriented, compact 4-manifold $M$. Is there a relation between the signatures ...
3
votes
1answer
32 views

The existence of a simply-connected neighborhood of a contractible loop

Let $M$ be a smooth manifold with a point $x_0$ on $M$ and a smooth loop $\gamma$ at $x_0$. If $\gamma=0$ in $\pi_1(M,x_0)$, then can we find a simply-connected open set $U$ around $x_0$ such that ...
1
vote
1answer
33 views

Derivation of the Maurer-Cartan formula

The left-invariant Maurer-Cartan forms are given by $$g^{-1}dg, $$ wher $g$ a Lie group $G$ to $M_n(\mathbb{R})$. My question is why is $$d(g^{-1}dg)=(g^{-1}dg)\wedge(g^{-1}dg)\quad ? $$ How come one ...
0
votes
0answers
32 views

$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
0
votes
1answer
24 views

inverse of stiefel-whitney class of product bundles

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a ...
1
vote
1answer
27 views

A Couple Formulas in Besse's “Einstein Manifolds”

In Besse's "Einstein Manifolds," Chapter 6D, there are 2 formulas which I am interested in, which apply to compact Riemannian $4$-manifolds: ...
0
votes
1answer
35 views

The signature of a product of surfaces

If $\Sigma_1$ and $\Sigma_2$ are surfaces (i.e. compact, oriented 2-manifolds without boundary), is the signature $\tau (\Sigma_1 \times \Sigma_2)$ well-known? Recall that the signature is the ...
0
votes
1answer
43 views

Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
2
votes
0answers
59 views

cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
1
vote
1answer
65 views

Is an configuration space contractible?

Let $\mathbb{R}^\infty$ be the union $\cup \mathbb{R}^n$(with inclusions). Let $F(\mathbb{R}^\infty,k)$ denote the $k$-th configuration space of $\mathbb{R}^\infty$, i.e., the subspace of ...
4
votes
1answer
53 views

map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
2
votes
0answers
42 views

Proving that homotopic maps have the same degree

Let $M, N$ be compact, connected, oriented manifolds. The degree of a map $f:M \rightarrow N$ is defined as the integer $k$ which satisfies $\int_{M} f^{*}\omega = k\int_{N}\omega$. Using the fact ...
1
vote
1answer
110 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
2
votes
1answer
49 views

Is $\mathbb{R}P^n$ “two-sided” in $\mathbb{R}P^{n+1}$

i.e. Does $\mathbb{R}P^n$ have a tubular neighborhood $N$ such that $N-\mathbb{R}P^n$ is disconnected. My guess is yes, but don't know how to show it convincingly ( or maybe only for $n$ odd, I'm ...
2
votes
3answers
300 views

Holomorphic functions on a complex compact manifold are only constants

Is there a simple proof that every holomorphic function $M\to\mathbb{C}$ on a compact complex manifold $M$ is constant?
2
votes
2answers
77 views

The meaning of $\omega$ = df?

I'm reading Algebraic Topology by William Fulton and I am a little lost about 1-forms, which I assume to be the number of differentials according to the number of dimensions (correct me if I'm wrong. ...
1
vote
1answer
39 views

Show this Map $f:M\rightarrow S^n$ is a Diffeomorphism

Let $M$ be a compact, connected smooth manifold of dimension $n>1$, and $f:M\rightarrow S^n$ a smooth map such that the differential $df_x:T_xM\rightarrow T_{f(x)}S^{n}$ has full rank $= n$ for all ...
0
votes
1answer
36 views

Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
3
votes
0answers
58 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
58 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
74 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
53 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
85 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
77 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
45 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
65 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
41 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
57 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
46 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
71 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
196 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
424 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 ...
6
votes
0answers
139 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
136 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
167 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
95 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
43 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
104 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
32 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
112 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
28 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
223 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
139 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 ...