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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wedge product and determinant

I don't really know what $[\phi_i(v_j)]$ really is. As far as I understand, $\phi_i$ is a linear transformation - a matrix; and $v_j$ is the column vector it eats. So $[\phi_i(v_j)]$ spit out column ...
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All 1-tensors are alternating

This statement from page 155 of Guillemin and Pollack's Differential Topology. I would assume because 1-tensors can not alternate because they have nothing to alternate with, so they are ...
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Which is harder to compute: $\pi_{n+k}$ or $\Omega^{fr}_n$?

Denote the $n+k$-th homotopy group of $S^n$ by $\pi_{n+k}(S^n)$ and the group of framed cobordism classes by $\Omega_n^{fr}(S^k)$. A central problem of algebraic topology is to compute $\pi_i(S^j)$ ...
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How is “index” at an Walrasian equilibrium proved? (in relation to Hopf-Poincare theorem)

So, the index of an (Walrasian/general equilibrium) equilibrium point is determined as the sign of $(-1)^{L-1} \times \det M$ where $M$ is a matrix and $M_{ij} = \frac{\partial{Z_i}}{\partial {p_j}}$, ...
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71 views

Regarding orientation and orientation-reversing in local diffeomorphism

I am confused about orientation and orientation reversing in local diffeomorphism $f$ from manifold $X$ to $Y$ at some points. So, what does $f$ orientation-reversing at a point mean?
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156 views

Relation between quadratic refinement and quadratic form

The question in the title has now been bothering me for days. I first came across the term quadratic refinement when I read about the Kervaire invariant when reading Kervaire's 1960 paper. The ...
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177 views

Importance of triangulation

Kervaire's seminal 1960 paper A manifold which does not admit any Differentiable Structure starts "An example of a triangulable closed manifold $M_0$ of dimension 10 will be constructed." What is the ...
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91 views

How does one prove that local diffeomorphism is submersion?

How does one prove that local diffeomorphism is submersion? For a manifold, what does it being disconnected mean? I get what "disconnected" means for a graph, but not for a manifold.
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Does $\mathrm{Mat}_{m \times n}$ have boundary?

To me, $\mathrm{Mat}_{m \times n}$ is isomorphic to $\mathbb{R}^{mn}$, hence is boundaryless. But this disqualified the use of Sard's theorem in this question: An exercise on Regular Value Theorem. ...
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77 views

Why $\Sigma$ is minimal, if $\frac{d}{dt} |_{t=0} \mathrm{Area}(\Sigma_t)=0$?

In this work http://arxiv.org/pdf/1204.2883v1.pdf Martin Li claimed that $\Sigma\subset M$ is minimal and $\Sigma$ meets $\partial M$ orthogonally along $\partial \Sigma$ if, only if, $$0 = ...
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312 views

A Surjective Local Smooth Diffeomorphism That is Not A Covering Map

Let $\pi:M_1\rightarrow M_2$ be a surjective $C^{\infty}$ map between two connected manifolds with $d\pi$ an isomorphism. If $M_1$ is compact, it is seen that $|\pi^{-1}(m_2)|$ is finite, so $\pi$ is ...
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44 views

Smooth immersion $\gamma$ from $\mathbb R \to \mathbb R^2$ such that $\operatorname{Im}\gamma=\{(x,\lvert x\rvert)\mid x\in \mathbb R\}$

Can I find a smooth immersion $\gamma$ from $\mathbb R \to \mathbb R^2$ such that $\operatorname{Im}\gamma=\{(x,\lvert x\rvert)\mid x\in \mathbb R\}$, I cannot take $\gamma(t)=(t,\lvert t\rvert)$ as ...
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358 views

An exercise on Regular Value Theorem

I got really stuck here for problem 2.3.8 on GP: Suppose $m > 1$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and let $K \subset \mathbb{R}^n$ be compact. Show that for any $\epsilon > 0$ ...
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116 views

A direct application of Sard's theorem

The question is let $f: X \rightarrow \mathbb{R}^2$, show that for almost every $c \in \mathbb{R}$, we have that $f^{-1}(\{c\}\times\mathbb{R})$ is a smooth submanifold of $X$. I want to apply Sard's ...
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170 views

Show that $dF_x$ is surjective for all $x$

I am trying to tackle question 2.3.8 on GP, but I haven't figure out the following question yet. Suppose $m > 1$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth map. Consider $f + Ax$ ...
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75 views

Showing that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is a smooth vector bundle over $S^1$

I want to show that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is the total space of a smooth vector bundle over $S^1$. Here $\sim$ is defined as follows: fix a linear isomorphism $L$ of ...
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273 views

Show that $f(C)$ has Hausdorff dimension at most zero.

We say that a set $A \subset \mathbb{R}^n$ has $d$-dimensional Hausdorff measure zero if for all $\epsilon > 0$ there exists a covering of $A$ by countably many cubes $S_i$ with side lengths $s_i$ ...
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24 views

Volum of the covering of $\bar{S} \geq S$?

The proposition on GP Page 203 says: Let $S$ be a rectangular solid and $S_1, S_2, \ldots$ a covering of its closure of $\bar{S}$ by other solids. Then $\sum$vol$(S_j) \geq$ vol($S$). This does not ...
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When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
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450 views

Prove that a standard torus is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$

I was asked to prove that a standard torus(which means we don't consider those pathological cases where it intersects with itself, e.g horn torus) is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$. ...
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154 views

Show that the tangent space of the diagonal is the diagonal of the product of tangent space

I'm stuck on this question for quite a few days and still haven't got a clue what to do. The question is as follows: If $\Delta$ is the diagonal of $X\times X$ where $X$ is a manifold, show that its ...
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61 views

Intuitive idea of tangent space and definition coincide?

Let $M$ be a submanifold of $\mathbb{R}^n$ of codimension 1. Suppose you take $V\le \mathbb{R}^n$ a vector space of dimension $n-1$ and let $w \in \mathbb{R}^n\setminus\{0\}$ be an orthogonal vector ...
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Tangent Space of Product Manifold

I was trying to prove the following statement(#9(a) in Guillemin & Pollack 1.2) but I couldn't make much progress. "Show that for any manifolds $X$ and $Y$, $$T_{(x,y)}(X\times Y)=T_x(X)\times ...
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188 views

Is a continuous map between smoothable manifolds always smoothable?

Let $X$ and $Y$ be topological manifolds and $f:X\to Y$ a continuous map. Suppose $X$ and $Y$ admit a differentiable structure (at least one). My question: is it always possible to choose a ...
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91 views

Find a manifold which contains embedding of $K_5$

$K_5$ graph is not planar . I was asked to find a manifold which contains embedding of $K_5$ and use $5$ squares to represent $K_5$ "on" my new manifold. Embedding means that it can be drawn on the ...
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46 views

Differentiability of $\operatorname{dist}(x,\partial \Omega)$ function [duplicate]

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with smooth boundary and set $$\phi(x)=\operatorname{dist} (x,\partial \Omega) =\inf_{y\in \partial \Omega} |x-y|$$ for $x\in ...
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70 views

Axiomatizing oriented cobordism

According to the nLab entry for abstract cobordism categories, the natural way of axiomatizing the relation of two oriented manifolds being cobordant is the following: Definition 1 Two objects ...
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Diagonal Inclusion Map of a manifold $X$

This question actually comes from the question I asked before: Derivative map of the diagonal inclusion map on manifolds And I repeat it as follows: Let $f: X\longrightarrow X\times X$ be the ...
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The “Easiest” non-smoothable manifold

In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
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183 views

Bounded vector field has globally defined flow

Let $X$ be a vector field on $\mathbb R^n$, and suppose that $\|X\|$ is bounded, where the norm is taken with respect to the Euclidean inner product. I am trying to show that $X$ has globally defined ...
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70 views

A proof on smooth function that I don't know what to proof.

Here's the question: Suppose $f: U \rightarrow V$ is a smooth map, for $U \subset R^k$ and $V \subset R^\ell$ open sets. That is, all partial derivatives (of all orders) of $f$ exist and are ...
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Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
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1answer
39 views

Prove closed of dimension one of $X\times I$.

Suppose $F(x,t): X\times I \rightarrow R$ is a homotopy of Morse functions. That is, $f_t: X \rightarrow R$ is Morse for every $t$. Show that the set $C = \{(x,t)\in X\times I : d(f_t)_x = 0\}$ forms ...
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Families of Morse functions

I don't have a clue with this problem. Thank you very much for your help & guidance. (a) Suppose $F(x,t): X\times I \rightarrow R$ is a homotopy of Morse functions. That is, $f_t: X \rightarrow ...
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124 views

Fubini Theorem for measure zero

I know Fubini Theorem in calculus, but the measure zero version does not make sense to me: $n=k+1$, and $V_c$ is the "vertical slice" {c}$\times R_l$. Let $A$ be a closed subset of $R^n$ such that $A ...
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41 views

Show the points $u,v,w$ are not collinear

Consider triples of points $u,v,w \in R^2$, which we may consider as single points $(u,v,w) \in R^6$. Show that for almost every $(u,v,w) \in R^6$, the points $u,v,w$ are not collinear. I think I ...
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176 views

Thom class: Why are the two definitions equivalent?

We know that the Thom class $\tau_W$ is defined on a disk bundle $W\rightarrow L$, where $L$ is a $p$-dimensional manifold and the rank of $W$ is $k$. Let $[W]_0$ denote the fundamental class of the ...
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81 views

0-manifold - final step

$f$ is a Lefschetz map on a compact manifold X. And I need to show the Lefschetz fixed point is isolated. I proved that the graph of f is transversal to the diagonal inside $X \times X$, then I don't ...
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1answer
106 views

Eigenvalue of f and df

Given 1 is not an eigenvalue of $df$ at $x_0$, take a chart $(U,\phi)$ around $x_0.$ Then in this coordinate neighborhood, think of $f$ as a map from open ball in $\mathbb{R}^n$ (say $B$), to itself ...
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Homomorphisms of Lie Groups

$SO(3)$ denotes the special orthogonal group, which is the (open) subset of $O(3)$ on which the determinant is one. I have shown that every element of $SO(3)$ fixes a line in $\mathbb{R}$ pointwise ...
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240 views

Finitely many Lefschetz fixed points

The questions is Show that if $X$ is compact and all fixed points of $X$ are Lefschetz, then $f$ has only finitely many fixed points. n.b. Let $f: X \rightarrow X$. We say $x$ is a fixed point of ...
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Approximating continuous functions $S^n \to S^n$

I'm trying to check that every continuous function $f:S^n \to S^n$ can be approximated by differentiable ones. Well, by Stone-Weierstrass I can approximate the coordinate functions $f_i:S^n \to \Bbb ...
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Injectivity of a map between manifolds

I'm learning the concepts of immersions at the moment. However I'm a bit confused when they define an immersion as a function $f: X\rightarrow Y$ where $X$ and $Y$ are manifolds with dim$X <$ ...
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If $f$ is a Morse function, then so is $f \circ \phi^{-1}$, where $\phi: U \rightarrow \mathbb{R}^k$ is the coordinate chart.

I am trying to show: if when $f^\prime = 0$, then $f^{\prime\prime} \neq 0 \Leftrightarrow (f \circ \phi^{-1})^\prime = 0$, $(f \circ \phi^{-1})^{\prime\prime} \neq 0$. But the problem is, because ...
3
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1answer
100 views

Morse Function Definition: Does it implies Morse function is $C^2$?

In my understanding, Morse function just means the determinant of Hessian matrix is nonsingular at critical points. So my claims are: the function itself should be continuous the reference to ...
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78 views

Prove that the tangent space of a hyperplane is itself

I know this might sound really stupid: I was trying to show that the tangent space of a hyperplane is itself. I started by parametrising the hyperplane locally at $x$ with a diffeomorphism $\phi : U ...
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75 views

Definition of diffeomorphism functions

I see the definition of Diffeomorphism in Wikipedia homepage, but I don't understand whether "the differential of f (Dfx : Rn → Rn) should be bijective at each point x in U" or "f itself" should be ...
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104 views

A basic proof on Morse Function

The questions is to show if $f_t$ is a homotopic family of functions 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 sufficiently small t. ...
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Fundamental Group, Piecewise Smooth Curves, Consevative Fields

Let M be compact Riemannian manifold. X be a vector field on M. I believe that work done by X any piecewise closed curve is zero, iff, the same is zero for a particular finite set of loops. I believe ...
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252 views

uniformization theorem - squares and circles

I am trying to understand the uniformization theorem and get some intuition about it. The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of ...