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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Integral of generator $H^{2k}(\mathbb CP^k)$ over $\mathbb CP^k$ is 1

I'm trying to give a proof of the Hirzebruch signature theorem from a differentiable viewpoint (i.e. using de Rham cohomology). The proof works, except for one small step. We know that ...
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(Locally) sym., homogenous spaces and space forms

We had some definitions of particular types of Riemannian manifolds in our lecture 1.) Locally symmetric spaces. They were Riemannian manifolds with the property that $\nabla R=0$ everywhere. 2.) ...
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resonance in pde

(I migrated the question from overflow b/c I think it belongs here instead.) About a year back I was in Professor Henry Mckean's office and he told me alot of interesting things. One thing he told me ...
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Mistake in the definitions of the linking number.

I am looking into the definition of the linking number. I've considered these two definitions. Consider a link $L$ with components $K_1$ and $K_2$, and respectively their embeddings $\gamma_1$ and ...
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30 views

Vanishing of the first Chern class of a complex vector bundle

Suppose that $E\to M$ is a $\mathbb{C}^n$-bundle with a metric. This is equivalent to saying that there exists a chart $\{U_\alpha\}$ of $M$ and $\phi_{\alpha,\beta} \colon U_\alpha\cap U_\beta\to ...
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19 views

Tilting sine function to get countably infinite nonregular values?

Let $f: \mathbb R \to \mathbb R$. A nonregular value $y$ of $f$ is any value such that not all $x \in f^{-1}(y)$ are regular. A point is regular if the Jacobian at it is surjective, in this case, has ...
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Why is this map $H^1$?

I have the following proposition (taken from Klingenberg's Lectures on Closed Geodesics): Let $\pi: E \rightarrow S$ and $\mathcal{O} \subset E$ be a finite dimensional fibre bundle over the ...
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Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that ...
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Lagrange's Equation on a Manifold

I know that, if $L: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}$, then the Euler-Lagrange equation is: $$ \nabla_x L - (\nabla_{\dot{x}}L)' \equiv 0$$ In trying to ...
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29 views

The fundamental vector fields of a principal bundle are vertical.

Let $p:P\to M$ be a principal $G$-bundle. To each $A$ in the Lie algebra of $G$ corresponds a fundamental vector field $A^*$ on $M$ defined by $$A^*_u=\frac{d}{dt}|_{t=0} u(exp(tA))$$ How can we see ...
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What can be said about the space of vector fields for which a given, say $C^1$, function is a Lyapunov function?

I am learning Morse homology and I have been thinking about the following observation. One way of doing, say finite-dimensional, Morse theory is by fixing a Morse function $f\in C^{\infty}(M)$, where ...
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89 views

Proving smoothness of left-invariant metric on a Lie Group

Assume $G$ is a Lie group. The standard construction of a left invariant metric on $G$ goes as follows: Take an arbitrary inner product $\langle,\rangle_e$ on $T_eG$ and define $\langle u , ...
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67 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
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190 views

Integrals of Pullbacks

This is a problem from Guillemin's Differential Topology: Suppose that $f_0, f_1: X \to Y$ are homotopic maps and that the compact boundaryless manifold $X$ has dimension $k$. Prove that for all ...
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51 views

About separation property of hypersurface

Let N be a complete Riemannian manifold and M be a complete hypersurface in N. M is said to have separation property if N\M is disjoint union of 2 connected open sets in N. Under what reasonable ...
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60 views

Existence of diffeomorphism through convergence in Hausdorff distance

I'm reading a book and have come across something that I cannot verify or fix. The assumption is that $\Omega_1, \Omega_2, ...$ is a sequence of connected open sets in $\mathbb{R}^n$ that converge in ...
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1answer
28 views

Tangent and normal spaces of submanifold of fixed-rank matrices

Let $m \geq 2$. The subset $X$ of $m \times 2$ matrices with rank $1$ is a (smooth) submanifold of $\mathbb{R}^{m\times 2}$. Let $A$ be in $X$. I know from a more general statement that the tangent ...
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33 views

Is it true that all $k$-submanifolds of a $m$-manifold are open subsets of some closed $k$-submanifold?

Let $M$ be a $m$-dimensional (smooth) manifold. I know that $m$-submanifolds of $M$ are exactly the open subsets of $M$. Is it true that all $k$-submanifolds of $M$ are open subsets of some closed ...
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A “parallel manifold” is always orientable

I want to solve the following problem from Spivak's Calculus on Manifolds: Let $M$ be an $(n-1)$ dimensional manifold in $\mathbb{R}^n$. Let $M(\varepsilon)$ be the set of end points of normal ...
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Locally ringed space locally isomorphic to a *closed* subset of $\mathbb{R}^n$

To me it's more natural to think of e.g. a tetrahedron as a closed subset of $\mathbb{R}^3$ than as a "manifold with corners" in the traditional sense -- i.e., locally isomorphic to open subsets of ...
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Incorrect statement in a proof of the transversality theorem?

I'm reading through Morris Hirsch's book on differential topology, and he makes the following offhand statement. Suppose k is a compact subset of a manifold U, and V is a vector subspace of R^n. If a ...
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1answer
54 views

Properly discontinuous action on manifold

I am actually not familiar with topology, but since we had a short outlook on these things in our differential geometry lecture today, I would appreciate some general remarks: Let $M$ be a smooth ...
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Local Submersion Theorem - Differential Topology of Guillemin and Pollack

Local Submersion Theorem : Suppose that $f:X \to Y$ is a submersion at $x$, and $y=f(x)$. Then there exist local coordinates around $x$ and $y$ such that $f(x_1,...,x_k)=(x_1,...,x_l)$. That is, $f$ ...
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Introductions to Ehresmann connections and Chern-Simons forms

I am looking for introductory texts on Ehresmann connections and Chern-Simons forms. I seek detailed, hands-on presentation. Please, recommend sources that employ a differential forms approach rather ...
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Relation beween transition functions of a principal fiber bundle and its dual

What is the relation between transition functions of a principal fiber bundle and its dual? As an example, consider the transition map of the frame bundle on a manifold of dimension $n$ as the ...
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39 views

Embedding covers of manifolds

I am considering $k$-fold covers of smooth manifolds (with smooth covering maps). Let $f:M^m\to N^m$ be a smooth finite covering map. -- The following implication is not true: $M$ can be embedded ...
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Homeomorphism between boundaries of open sets is “surjective”

Let $U,V\subset\mathbb{R}^n$ two bounded open sets and let $F:\overline{U}\to\mathbb{R}^n$ be a continuous map. Assume that $F$ maps $\partial U$ homeomorphically onto $\partial V$. The question: is ...
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Need help with this proof, I don't understand it , could anyone clarify some of the details. System of linear Differential equations.

$$(*)X'=A(t)X - system$$ $$(*)PX(\alpha)+QX(\beta)=0.$$-border conditions, where P,Q constant square matrices $n \times n $. Let $Y(t)$ be the fundamental matrix for the system $(*)$ normed for$ t= ...
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Definition by degree and intersection number are equivalent (linking number). [repost]

I will here restate a question I asked earlier. It did not have much succes (probably by an incomplete introduction of the problem on my part). I am reading a paper by Ricca ( ...
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Application of Morse Inequalities

I am an undergraduate student interested in morse theory. I understand, that the morse inequalities provide an lower bound for the number of critical points morse functions on a manifold can take. One ...
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43 views

Quotient manifold theorem provides a fibration?

It's known that if $G$ is a Lie group and $H\subseteq G$ is a closed subgroup, then the quotient map $p\colon G\to G/H$ is in fact a principal $H$-bundle, which follows from the existence of local ...
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2answers
63 views

Lie group step in proof

Let $X_e,Y_e \in T_eG$ be vectors and $G = GL(n).$ Then the right translation is given by $Y_g = Y_e g$ and $X_g = X_e g.$ Now, I have a proof showing that $[X_e,Y_e] \in T_eG$ is the element ...
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How does Morse theory on non-compact manifolds differ from compact manifolds?

What is the Morse homology of a non-compact manifold? When is it, as in the compact case, isomorphic to singular homology of the underlying manifold? What other constructions can be identified with ...
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1answer
45 views

underlying real vector bundle of a complex vector bundle

Let $\eta^\mathbb{C}$ be a complex line bundle. If the underlying $2$-dimensional vector bundle $\eta$ is not trivial as a real vector bundle, can we obtain that $\eta^\mathbb{C}$ is not trivial as a ...
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1answer
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Construct a $k$-form on $S^k$ with nonzero integral

How to construct a $k$-form on $S^k$ with nonzero integral ? I think this can be done by Bump function $\rho$ on $R^k$ and define $w = \rho\, dx_1\,dx_2 \cdots dx_k$ on $R^k$. Now pull it back by the ...
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1answer
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Lemma about Brouwer degree in Milnor's book

In Milnor's book "Topology from the Differentiable Viewpoint", He states the following lemma: Let $M, N$ be oriented $n$ manifolds, with $M$ compact and $N$ connected. Let $f : M \to N$ be a smooth ...
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1answer
22 views

Orientability of Orbit Space of a Group of Diffeomorphism

While working on some problem in differential topology, I had to prove the lemma below. It seems to me like I have not needed all the requirements in the lemma, leading me to think that I have ...
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1answer
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If $F:M\to N$ is a smooth embedding, then so is $dF:TM\to TN$.

Question: I am trying to show that if $M$ and $N$ are smooth manifolds (without boundary), and $$F:M\to N$$ is a smooth embedding, then the differential $$dF:TM\to TN,\quad ...
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Whether or not such a simple CW complex can be made a $C^{\infty}$ manifold?

Problem Let $X$ be the space obtained by attaching two disks to $S^1$, the first disc being attached by the 7 times around,i.e. $z \to z^7$, and the second by the 5 times around. Can $X$ be made ...
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What does it mean for two submanifolds to have contact of order $k$?

Let $M$ be a smooth manifold of dimension $(m+n)$. Two curves $\gamma_1, \gamma_2 \colon \mathbf{R} \to M$ with $\gamma(0) = p$ are said to have contact at $p$ of order $k$ if for all smooth maps ...
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76 views

Relations on Stiefel-Whitney classes

Can arbitrary cohomology classes $w_1,\dots,w_n$ from $H^{*}(B,\mathbb{Z}_2)$ be Stiefel-Whitney classes of some bundle over the given base $B$ or there are some necessary relation which can be ...
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Differential equations, theoretical question

$$x(t)={1 \over (c-t)}\tag{$**$}$$is the general solution of $$ x'=x^2$$ Why is the area existence and uniqueness of the solution $R_{tx}^2??$ Its the general solution in : $G_1=\{(t,x): t\in R , ...
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Definition of parameterised $n$-manifold

Let $U \subseteq \mathbb R^{n+q}$ and let $W = U \cap \mathbb R^n \times \{0\}$. Let $\phi : U \to \phi (U)$ be a diffeomorphism. Then $M=\phi (W)$ is called a parameterised $n$-manifold in $\mathbb ...
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1answer
37 views

Cobordism: Reference Request.

Where does one learn Cobordism theory - is there some canonical text or reference at the beginning graduate level? More general sources on modern differential topology (surgery etc.) are fine; any ...
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Taylor Expansion of tensor moved by a flow.

I am reading Peter Petersen's notes on manifold theory and he introduces Lie Derivatives in the following way. "Let $X$ be a vector field and $F^t$ the corresponding locally defined flow on a smooth ...
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When does $\pi_1(\Sigma)$ inject into $\pi_1(S^3 \setminus \Sigma)$?

Here's a fun fact from knot theory: $\quad$ If $\, \Sigma$ is a minimal-genus Seifert surface for a knot $K$, then $i_*:\pi_1(S^3 \setminus \Sigma) \to \pi_1(S^3 \setminus K)$ is injective, where ...
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1answer
50 views

Is there a one to one correspondence between Jones' polynomials and knots?

I know Jones' polynomial is a knot invariant. By using knot invariant like p-coloration one can only say whether two knots are different but not whether they are the same. So it is like injective ...
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Why is this map called a fold?

Consider the map $\varphi : \mathbb R^2 \to \mathbb R^2$ defined by $(x,y) \mapsto (x,y^2)$. Apparently this map is called a fold as the $(x,y)$-plane is folded over and creased along the axis ...
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Zips and Zippers

I'm currently reading Differential Manifolds by Antoni Kosinski, and the concept of a zip--defined as half of a zipper--is mentioned early on, of course with the intent of connecting manifolds. This ...
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Proving that Lie groups are parallellizable

Let $G$ be a Lie group. There is a diffeomorphism $$G \times T_e G \to TG$$ mapping $(g, [\gamma]) \mapsto [g \cdot \gamma]$. The inverse map then gives rise to the following isomorphism of bundles: ...