Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Tangent bundles of smooth manifolds

Using the identity $T(M \times N) = T(M) \times T(N)$, it is easy to construct the tangent bundles for various smooth manifolds such as the n-dimensional sphere $S^{n}$. However, I could not figure ...
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Operations on a smooth vector bundle

On a smooth vector bundle, one often defines addition and scalar multiplication to form a vector space. However, doesn't one need to show that these operations are smooth? Is this trivial or is there ...
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Topology of the intersection of toric arrangement

Hope someone will help me in the solution of the following question. I'm working on some topological problem involving the topology of the intersection of some characters of the torus. I want ot find ...
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Existence of transversal intersection for $M$ submanifold and of some hyperplane.

Let $M^n\subset\mathbb{R}^P$ submanifold, show that there exist a hyperplane $H^{p-1}$ in $\mathbb{R}^P$ sucht that $H^{p-1}$ intersect $M^n$ tranversally. This problem is I prove using the next: ...
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177 views

Whitney sum of smooth vector bundles

I was reading through Lee's smooth manifolds book, in his chapter on vector bundles. Upon reading about smooth vector bundles and its definition, I was wondering if the whitney sum of two smooth ...
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49 views

Let $\pi: E \to M$ a vector bundle. Is $E$ a direct summnad of $M\times\mathbb{R}^{d}$, for some $d$?

Let $\pi: E \to M$ a vector bundle over a smooth manifold $M$. $E$ is direct summand of $M\times\mathbb{R}^{d}$, if there exist a vector bundle morphisms $f:E\to M\times\mathbb{R}^{d}$ and $h:M\...
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53 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, \begin{...
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Definition of topological manifold

This might be a stupid question, but I was wondering why we define the topological manifold to be Hausdorff and Second countable? Thanks :)
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181 views

Partial derivatives on Manifolds

Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$ $E_2 F: A ...
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173 views

Show every $f_t$ is Morse for $t$ is sufficiently small

Let $f$ be a Morse function on the compact manifold $X$. Let $f_t$ is a homotopic family function with $f_0=f$. Show every $f_t$ is Morse for $t$ is sufficiently small Here is my argument, but my ...
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170 views

Find a map of the solid torus into itself having no fixed point. Where does the proof of the Brouwer theorem fail.

Find a map of the solid torus into itself having no fixed point. Where does the proof of the Brouwer theorem fail. I know that the proof is fail because the torus has a hole, so we can't construct ...
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119 views

Prove that the Brouwer theorem is false for the open ball $|x|^2 <a$

Prove that the Brouwer theorem is false for the open ball $|x|^2 <a$ Brouwer Theorem: Any smooth map $f$ of the close unit ball $B^n \subset R^n$ tin to it self must have a fixed point. I need to ...
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1answer
41 views

Prove isomorphism of fundamental groups

Hei, guys! I'm having some problem solving the next exercise: Let $f: M -> N$ be a homeomorphism. Define a map $f*:π_1 (M, x_0) → π_1 (N, f(x_0 ))$ such that $f*([\gamma])=[f∘\gamma]$. Show that $...
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91 views

Prove that$ H_x (X)$ does not depend on the choice of local parametrization.

Suppose that $X$ is a manifold with boundary and $x∈∂X$. Let $ϕ:U→X$ be a local parametrization with $ϕ (0)=x$ where $U$ is an open subset of $H^k$. Then $dϕ_0:R^k→T_x (X)$ is an isomorphism. Define ...
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66 views

Hausdorff property of $\mathbb{RP}^n$ from unusual definition

Rather than defining the topology on $\mathbb{RP}^n$ as the quotient $(\mathbb{R}^{n+1}\backslash\{0\})/$~ or $S^n/$~ in the usual way, suppose you use these equivalence relations simply to define a ...
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75 views

Canonical projection on submanifold $M^k $ over a hyperplane $H^{n}$ is immersion

Let $M^k \subset \mathbb{R}^{n+1}$, $M$ compact and $2k\leq n$. Show tha exist a $n$-hyperplane $H^n\subset \mathbb{R}^{n+1}$ such that if $\pi:H^{n}\oplus (H^n)^{\perp}\rightarrow H^{n}$ is the ...
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1answer
127 views

Apply the theorem of tubular neighborhood

$M$ is a connected manifold, $N\subset M$ is a connected submanifold with nontrivial normal bundle, and dimM-dimN=1. How to prove $M-N$ is connected? There is a hint to use the tubular neighborhood ...
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118 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'\\ ...
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1answer
59 views

Can a smooth function $f\colon\partial D^n\to\partial D^n$ be extended to a smooth function $\hat{f}\colon D^n\to D^n$?

Suppose for $n\geq 1$, you have a smooth map $f\colon S^{n-1}\to S^{n-1}$. Viewing $S^{n-1}=\partial D^n$, is it possible to extend $f$ to a smooth map $\hat{f}\colon D^n\to D^n$, $D^n$ being the ...
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76 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 (http://en.wikipedia.org/wiki/Signature_(...
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71 views

Is cone not a topological manifold?

Is the cone = X a Hausdorff, second-countable topological space that is not a topological manifold? Since the open subsets $U_{\alpha}$ do not cover the vertex of the cone, so $U_{\alpha}$ is not a ...
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83 views

Submersion and immersion

I googled wiki about submersion and immersion. Wiki states that submersion is dual to immersion. I wonder where this duality relationship comes from.
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318 views

The intuition of the rank theorem

In Rudin's Principle of Math Analysis, I know the proof of the rank theorem. However, I fail to understand its content. 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ ...
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224 views

Finding bump function on a smooth manifold using partitions of unity.

Let $M$ be a smooth manifold. Let $A$ and $B$ be disjoint closed sets of $M$. Show there exists a smooth function $f$ such that $f^{-1}(0)=A$ and $f^{-1}(1)=B$. This is my idea so far, Since $A$ ...
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122 views

Exponential map on Diffeomorphism group of $S^1$

I am reading Segal book on Loop groups, and he mentions the following theorem: $$ \exp: Vect(S ^1) \rightarrow Diff(S ^1) $$ the map taking a vector field to the diffeomorphism obtained by flowing ...
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1answer
170 views

A question about contractibility of a Lie group

In Lawson's book Spin Geometry, chapter II, before Proposition 1.11, it mentions a fact that if the 1st, 2nd and 3rd homotopy groups of a Lie group $G$ vanish (although $\pi_2(G)=0$ automatically), ...
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1answer
124 views

Prove that $f$ map a neighborhood of $Z$ diffeomorphically on to a neighborhood of $f(Z)$

Supposed that the derivative of $f:X\to Y$ is an isomorphism whenever $x$ lies in the sub-manifold $Z \subset X$, and assume that $f$ maps $Z$ diffeomorphically onto a $f(Z)$ . Prove that $f$ map a ...
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De Rham Cohomology is Sheaf Cohomology

I'm trying to prove that de Rham cohomology computes the cohomology of the constant sheaf $\mathbb{R}$ of real valued smooth functions on a manifold. I would like to do this without using Čech ...
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Prove that if $S^k$ has a non vanishing vector field, then its antipodal map is homotopy to the identity if $k$ is even.

Prove that if $S^k$ has a non vanishing vector field, then its antipodal map is homotopy to the identity if $k$ is even. Here is what I got so far. Suppose we have an antipodal map $x \to -x$ of $S^...
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Prove this plane algeraic curve is not a differentiable manifold

Prove the algebraic curve $\{(x,y)~|~x^2(x+1)-y^2=0\}$ in $\mathbb{R}^2$ is not a differentiable manifold. Remark: It is evident that the given cubic curve has a singularity at $(0,0)$ which disable ...
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103 views

Show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for $t$ is sufficiently small.

Suppose that $f_t$ is a homotopic family of function on $R^k$. Show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for $t$ is sufficiently small. I know that ...
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90 views

Prove that $f$ is Morse function if an only if $det(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$

Let $f$ be a smooth function on an open set $U\subset R^k$. For each $x \in U$ let $H(x)$ be the Hessian Matrix of $f$, whether $x$ is critical point or not. Prove that $f$ is Morse function if an ...
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94 views

Existence of smooth diffeomorphism $f$ of open ball onto itself with $f(0) = p$.

I am trying to show that for every point $p$ of the open $n$-disk $B^n$, there exists a smooth diffeomorphism $B^n \to B^n$ sending $0$ to $p$. Certainly, it seems intuitively obvious for points $p$ ...
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Naturality of the pullback connection

I'm completely stuck proving the naturality of the pullback connection. The strategy suggested is a follows: We let $\phi: (M,g) \to (\tilde{M}, \tilde{g})$ be an isometry, with connections $\...
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68 views

Noncomplete riemannian manifolds

In Lee's Riemannian manifolds text, he claims that "on $\mathbb{R}^n$ with metric $(\sigma^{-1})^* g$ obtained from the sphere from stereographic projection, there are geodesics that escape to ...
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1answer
196 views

Divergence in terms of Levi-Civita connection

The divergence of a vector field $X$ on a manifold $M$ is defined usually as the function $\text{Div}(.)$ such that $(\text{Div} X) \;\mu =L_X \mu$ for $\mu$ a volume form. I know that there is also ...
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56 views

The derivative as a linear function

In Milnor's Topology From the Differentiable Viewpoint, the derivative of a smooth map $f: U\to V$ is defined as $$ \mathrm{d}f_x: \mathbb{R}^k \to \mathbb{R}^l $$ $$ h\mapsto \lim_{t\to0} \frac{f(x+...
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Diffeomorphism between $\mathbb{R}^{2}/\sim$ (Torus ) and $\mathbb{S}_{1}\times \mathbb{S}_{1}$

This is homework so no answers please. Consider the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$ ,where $(x_{1}y_{1})\sim (x_{2},y_{2})$ iff $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$. I ...
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520 views

Tangent bundle to 2-sphere isnt trivial as a vector bundle?

I read many quiestion about $TS^{2}\ncong S^{2}\times\mathbb{R}^{2}$ where the hint is use Hairy ball theorem and directly is done. My question is: how do I proof that $TS^{2}\ncong S^{2}\times\...
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Calculate Brieskorn Manifold? [duplicate]

I need show that Brieskorn Manifold is submanifold with dimension $2n-1$ and calculate specifically for $d=2$ and $n=1$ $W(d)=\lbrace (z_{0},z_{1},...,z_{n})\in \mathbb{C}^{n+1}\vert$ $ z_{0}^{d}+z_{...
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183 views

A question regarding the proof of Hopf's theorem.

This is a question regarding the proof of the Hopf theorem given in "Topology from a Differential Viewpoint" by Milnor: If $v:X\to \Bbb{R}^m$ is a smooth vector field with isolated zeroes, and if $...
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1answer
98 views

Equivalence between orientation of the tangent bundle and orientation of manifolds

If $M^{n}$ is a manifold then the following statement are equivalent. The tangent bundle $(TM,\pi,M)$ is an orientable $n$-dimensional vector bundle. $M$ has an $\lbrace (U,h)\rbrace$ atlas on $M$ ...
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120 views

Intuitive idea of immersions?

So I understand the definition of immersions and submersions, as well as the motivation for defining such ideas. Not only are they important in understanding properties of mappings of tangent spaces, ...
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134 views

why is there no non-degenerate 2-forms on 4-sphere?

The question is in the title. I have been told that there are actually no non-degenerate 2-forms on $S^{2n}$ for $n \neq 1,3$. I have found the following question: No symplectic structure on $S^{2n},...
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268 views

A question from Milnor's “Topology from a differentiable viewpoint”

Milnor's "Topology from a Differentiable Viewpoint" says the following: Let $f:M\to N$ be a smooth mapping, where $M$ is $m$ dimensional and $N$ is $n$ dimensional. Moreover, $m\geq n$. If $y\in N$...
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1answer
48 views

The differential of a smooth map on manifold at points of local maxima

I have a differentiable function $f:M \to \mathbb{R}$ where $M$ is a smooth manifold. If $p \in M $ is a point of local maxima, that is I have an open set $V \subset M$, $p \in V$, so that $f(p)\geq f(...
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49 views

Invariant forms on principal bundles

Let $\pi:M \to B$ be a principal $G$-bundle and $\xi$ a invarint $k$-form on $M$. Does $k> dimG$ implies that $\xi$ is a basic form (pull back of a $k$-form on the base manifold $B$)?
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Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?

Why does $\chi(SL_n(R))=0$? I'm going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper ...
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50 views

Given points $x,y$, and tangent vectors $v,w$, is there a diffeomorphism $x\mapsto y$ and $v\mapsto w$?

Suppose $M$ is a smooth manifold, and you're given two points $x,y$, and associated tangent vectors $v$ and $w$. I'm curious, does there exist a diffeomorphism $F$ such that $F(x)=y$ and $dF_x(v)=w$? ...
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80 views

Representative of a cohomology class in once punctured solid torus

Consider a once punctured solid torus $(\mathbb R^2 \times S^1) /\{pt\}$. It is not difficult to see that it is homotopy equivalent to the bouquet of spheres $S^2\vee S^1$. So this guy has a non-...