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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$\{\gamma \in C^0([0,1],M): \gamma (0)=p, \gamma (1)=q\}$ with the compact-open top has the homotopy type of a CW complex

...where $M$ is a smooth manifold and $p, q \in M$. Does anyone know of any slick or accessible proofs of this? I was referred to Milnor's "On Spaces Having the Homotopy Type of a CW Complex" which ...
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Action of U(1) on a sphere bundle

Suppose $N$ is a closed, $n$-dimensional, orientable smooth manifold. Moreover, $n$ is odd. Consider the tangent bundle $TN$ of $N$. By adding a point at infinity to each tangent space I define a ...
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Non-intersecting smooth paths in the plane and the relation to curvature

I'm interested in a problem and have no idea how to approach solving it. Could you please point me in the right direction. Given 3 smooth paths $\varphi_{1}:[0,1]\to \mathbb{R}^{2}$, ...
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What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface?

Let $X$ be a compact Riemann surface of genus $g>1$, $f\in Aut(X)$, a biholomorphism of $X$ onto itself, $x\in X$ a fixed point of $f$. Since tangent map of a holomorphic map (on the real tangent ...
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“Immediate” Applications of Differential Geometry

My professor asked us to find and make a list of things/facts from real life which have a differential geometry interpretation or justification. One example is this older question of mine. Another ...
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Diffeomorphism group of the unit circle

I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and ...
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Real analytic diffeomorphisms of the disk

Is there any real analytic diffeomorphism from two dimensional disk to itself, except to the identity, such that whose restriction to the boundary is identity?
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extension to the ball of sphere immersion

What are the constraints to extend an immersion of the sphere $S^2$ into $\mathbb{R^3}$ to an immersion of the closed unit ball $B(0,1)$ to $\mathbb{R}^3$? Suppose, I get an immersion of $S^2$ into ...
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Properties preserved by diffeomorphisms but not by homeomorphisms

Diffeomorphisms between manifolds are particular homeomorphisms, so each property preserved by homeomorphisms is preserved by diffeomorphisms. Can you show me some examples of properties preserved by ...
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metric on the Euclidean Group

I am not an expert in this so I hope this doesn't sound so stupid: what is the common metric used when studying the Euclidean Group $\mathrm{E}(3)$. One could probably ask the same thing for ...
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Higher dimensional Jordan-Brower separation theorem

Is there an established higher dimensional version of the Jordan-Brower theorem? I would like some statement like this: Let $M$ be an $n$-dimensional closed, compact and connected variety and suppose ...
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Open subsets in a manifold as submanifold of the same dimension?

An open set in an $n$-manifold is clearly a submanifold of the same dimension as its containing manifold (see open manifolds). Now, given an $n$-manifold $M$, is it true that a set, to be the ...
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The embedding of manifold

If $M$ is a smooth manifold and $f$ is a smooth function on it, is it necessarily that there is an embedding $F$ of $M$ to $\mathbb R^k$ such that the first component of $F$ is $f$ ?
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Holomorphic immersion

Is it possible to get an holomorphic immersion form $\mathbb{C}$ to $\hat{\mathbb{C}}$ which is surjective? Here immersion means that $f'(z)\neq 0$ when $f(z)$ is finite and ${\left(\frac ...
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Is the Euler Number of A Vector Bundle Always Finite?

Let $E$ be an oriented vector bundle with an even rank $n$ over a smooth oriented $n$-dimensional manifold $M$, $e(E)$ denotes the corresponding Euler class, then the Euler number is defined by ...
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Can one average “close” smooth functions?

Suppose $M$ is a connected, smooth, second-countable manifold. Let $U \subset M^n$ be some neighbourhood of the diagonal. We will call a function $a: U \times \Delta_n \rightarrow M$ an "averaging ...
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Poincaré Duality with de Rham Cohomology

This is probably a fairly basic question: Poincaré Duality states the following: Given an $n$-manifold, the $k^{th}$ homology is isomorphic to the $(n-k)^{th}$ cohomology. So I was curious is there ...
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Compact submanifold represents a homology class

I've heard somewhere that a compact submanifold of a manifold gives a homology class of the whole manifold and vice versa. Can somebody explain this to me?
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What's the shining sparkle in the Sphere Inside-Out problem?

I've just seen the wonderfully done movie Sphere Inside Out, one about the Smale's paradox. And the first question came in mind is that, why it has to be so ugly? Why turning an ultimately simple ...
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Diffeomorphisms of $\mathbb{R}^n$

I recently realized that I don't know any non-linear diffeomorphisms of the plane (or $\mathbb{R}^n$ in general) except for linear ones, so I want to ask rather broad questions hoping to be pointed to ...
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How to construct a vector field without zero on an open manifold?

a friend asked me to pose the following problem: It is known that on an open manifold (connected, not compact and without boundary) there exists a vector field without zero, since its Euler ...
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Ambient Isotopy

From Hirsch's Differential Topology, p. 180. The first of the isotopy extension theorems says; Let $A\subset M$ be a compact submanifold and $F:V\times I \rightarrow S^{3}$ an isotopy of $A$. If ...
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Question about the proof of the index theorem appearing in Milnor's Morse Theory

I am trying to get through the proof of the index theorem. The background: I have been stuck for quite a while on the following point which Milnor says is evident: Let $\gamma: [0,1]\rightarrow M$ be ...
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Locally flat submanifold

Recently I found the following definition: Let $M^{n}$ be an $n-$dimensional topological manifold. Then $N^{k}\subseteq M^{n}$ is a locally flat submanifold if for every $x\in N$ there exists an open ...
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Proof that vector area is a boundary integral?

Let $M \subset \mathbb{R}^2$ be a closed topological disk and let $f: M \rightarrow \mathbb{R}^3$ be a smooth embedding; let $N$ be the corresponding unit normal field on $M$. The vector area is ...
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Are all vector bundles “flat vector bundles”?

This concept appears in Bott&Tu's GTM82. A flat vector bundle is one who has a particular trivialization with locally constant transition functions. Then my question is whether every vector bundle ...
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equivalence between differentiability definitions

In analysis course we encounter commonly the following definition of differentiable function: $f:U \rightarrow \mathbb{R^m}$, where $U \subset \mathbb{R^n}$ is differentiable when $\exists \ T \in ...
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An example of a derivation at a point on a $C^k$-manifold which is not a tangent vector

Let $M$ be a real $C^k$-manifold, $\mathscr{H}_x$ be the $\mathbb{R}$-algebra of germs of $C^k$-differentiable functions at $x \in M$. It is easy to show that $T_x M$ can be embedded into the space of ...
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Is a vector bundle orientable if and only if its dual bundle is orientable?

I was reading up on my dual spaces today and I made the following hypothesis: A vector bundle $\xi$ is orientable if and only if $\xi^*$ is orientable. This seems rather intuitive, and although ...
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Reversing the Ricci flow

Suppose $S$ is a closed, oriented surface (2-manifold) embedded in $\mathbb{R}^3$, which inherits the metric from $\mathbb{R}^3$, so that distances are measured by shortest paths on the surface. If ...
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Obstructions to lifting a map for the Hopf fibration

This is a bit of an elementary question, but suppose $\pi: \mathbb{S}^3\to \mathbb{S}^2$ is the Hopf fibration, are there reasonably computable obstructions to when a map $f:M\to \mathbb{S}^2$ can be ...
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A question about complex manifolds

Let $(M,J_{M})$ be a almost complex manifold and $(N,J_{N})$ be a complex manifold. I want to prove that $F^{*}(\mathcal{O}_{N})\subset\mathcal{O}_{M}$ implies that $F:M\rightarrow N$ is almost ...
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What does “locally trivial” do for us?

For the following we will work in the smooth category. (But examples in the topological category is also welcome.) The usual definition of a fibre bundle is Def A fibre bundle is the quadruple ...
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basic differential forms

Given a fiber bundle $f: E\rightarrow M$ with connected fibers we call the image $f^*(\Omega^k(M))\subset \Omega^k(E)$ the subspace of basic forms. Clearly, for any vertical vector field $X$ on $E$ we ...
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An exercise in Spivak's *Calculus on Manifolds*

Problem 5-6 in Michael Spivak's Calculus on Manifolds reads: If $f:\mathbb R^n\to\mathbb R^m$, the graph of $f$ is $\{(x,y):y=f(x)\}$. Show that the graph of $f$ is an $n$-dimensional manifold if ...
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Are there bounded surfaces without boundary that are noncompact?

I am aware of the Heine-Borel theorem, which says that closed and bounded is the same as compact in $\mathbb{R}^n$. My question is: are there (connected) surfaces without boundary embedded in ...
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Equivalent definitions of differential map

Let $f:M\rightarrow N$ be a smooth map between smooth manifolds, let $p\in M$ and $v\in T_{p}M$. Two different definitions of differential maps on tangent space: let $\gamma$ be a smooth curve on $M$ ...
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Useful topology on space of smooth structures on $\mathbb R^4$?

Mathoverflow is intimidating, so I thought I'd ask here first (second). If I don't get any useful answers here in a few days, I'll ask there. $Q_0$: Is there any use for a topology on the (continuum ...
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Embedding a manifold in the disk

I don't understand a sentence made by Hirsch in his Differential Topology at page 175: If $k > n+1$ and $M^n = \partial W^{n+1}$, then an embedding $M^n \hookrightarrow S^{n+k}$ extends to a neat ...
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Is $\mathbb{S}^\infty$ exotic

During construction of universal bundles one considers (for example) the infinite real projective space $\mathbb{R}\mathbb{P}^\infty$, coming from the sphere $\mathbb{S}^\infty$. My question is, are ...
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Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory

I am looking for a textbook that might serve as an introduction to principal bundles, curvature forms and characteristic classes, and perhaps towards 4-manifolds and gauge theory. Currently, the only ...
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Construction of Exotic Spheres

Milnor was constructing exotic spheres (at least in dimension 7) by bundle theory. Having proven the existence of such an exotic beast, I wonder if something as this is possible: Let $\mathbb{S}^n$ ...
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576 views

The definition of submanifold of a manifold with boundary

What is the exact definition of submanifold of a manifold with boundary? For example, when $H$ is the half space of the plane and S is a cycle which intersects with the origin in the half plane. Then ...
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Definition of Cobordism

I am currently trying to read Quillen's Cobordism paper - On the formal group laws of unoriented and complex cobordism theory Bull. Amer. Math. Soc, 75 (1969) pp. 1293–1298. Still on the first page ...
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What is wrong with this proof (that all vector bundles of the same rank are isomorphic)?

Suppose I have two vector bundles $E \rightarrow M, E' \rightarrow M$ of rank $k$ on a smooth manifold $M$. Let $\mathcal{E}(M), \mathcal{E'}(M)$ denote their spaces of smooth sections. We can ...
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Intuition for smooth manifolds

Consider the graphs of the functions $f_1(x) = |x|$, and $f_2(x) = x$ under the subspace topology of $\mathbb{R}^2$. Both of these graphs are smooth manifolds, just pick coordinate charts to be $(x, ...
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Confusion about Poincaré-Hopf

The two following theorems appear to be contradictory. Both are proved in Milnor's "Topology from the differentiable viewpoint." I'm sure I'm overlooking something incredibly trivial: Let $f$ be a ...
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Compute the degree of map

Let $S:SU(2)\rightarrow SU(2)$ be defined as $S(X)=X^{4}$. Compute the degree of $S$. Now, $SU(2)$ is homeomorphic to $S^{3}$, so the degree can be taken as:$$ \int_{S^{3}}S^{*}\omega=(\deg ...
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Degree of Gauss map equal to half the Euler characteristic and Poincaré-Hopf

The Poincaré-Hopf theorem states that for a smooth compact $m$-manifold $M$ without boundary and a vector field $X\in\operatorname{Vect}(M)$ of $M$ with only isolated zeroes we have the equality ...
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Books on topology and geometry of Grassmannians

Grassmannians are a pretty useful subject in numerous fields of mathematics (and physics). In fact, it was the first non-trivial higher-dimensional example that was given in an introductory projective ...