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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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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185 views

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 ...
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Software to draw links or knots

I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots. What I am looking for is the ...
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262 views

Why do we integrate 1-forms?

So integration of a 1-form $\omega$ over a path $\gamma$ is defined to be the integral of the pullback of $\omega$. Why does this make sense? Why don't we integrate over a vector field instead, like ...
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449 views

Understanding differential form

Let $M$ be a smooth manifold. A differential form of degree $k$ is a smooth section of the $k$th exterior power of the cotangent bundle of $M$. Does it mean that a differential form of degree ...
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166 views

Tangent bundle of $\mathbb{RP}^n$

I am trying to show that $T\mathbb{RP}^n$ and $\text{Hom}(\gamma_1,\gamma_1^{\perp})$ are isomorphic bundles over $\mathbb{RP}^n$. For $[x]\in\mathbb{RP}^n$, let $L_x$ be the line in ...
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235 views

Algebraic Link/Knot not of the Torus Type

I'm studying Milnor's Singularities of Complex Hypersurfaces, and a small, perhaps moot, point in Chapter 10 has me thinking in circles. (I asked a related but different question here). Here is some ...
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322 views

De Rham Cohomology Question

Let $M$ be a compact connected $n$-manifold without boundary. Let $\mu\in\Omega^{n-1}(M)$, show that there exists a point $p\in M$ such that $d\mu(p)=0$.
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141 views

Is there such kind of theorem saying two homotopic ways of attaching handle result in diffeomorphic manifolds?

M' is a manifold with boundary. One can attaching a handle $h:=D^k\times D^{n-k}$ along $f:S^{k-1}\times D^{n-k}\rightarrow M'$ forming $M=M'\cup_f h$. Suppose f is homotopic with f', and that ...
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Smooth structure on the topological space

Consider a topological space $X$. Lee in Introduction to Smooth Manifolds wrote that it is impossible to introduce a smooth structure on the topological manifold based only on topology (i.e. ...
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131 views

Relationship between integrable 1-plane fields and a one parameter group of diffeomorphisms

A 1-plane field on a manifold is a smooth choice of a 1 dimensional subspace of the tangent space at every point. If this plane field is integrable then there is an associated 1-dimensional foliation ...
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How to prove that the complex in Morse homology is isomorphic to the one in cellular homology

Since every stable submanifold with orientation in Morse homology is actually a cell in cellular homology, it suffices to prove the two boundary map coincide. Intuitively one may accept it is true by ...
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why not just 2 charts to make atlas for sphere?

In http://en.wikipedia.org/wiki/Manifold_(mathematics)#Construction, it says that 6 charts can be used to make an atlas for a sphere. But the text shows that you have a chart for the northern ...
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145 views

Properties of Surgery on Manifolds?

I am trying to give a brief explanation in which I make use of the concept of surgery on an $m$-manifold $M$. This is along the lines of (and taken generously from) the Wikipedia entry on Surgery; ...
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154 views

Topology of pseudo projective space

I don't know if "pseudo projective space" is a general accepted term, but I once read a book on general topology where the term was used for $\mathbb{S}^n / (\mathbb{Z}/m\mathbb{Z})$ (where you get ...
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200 views

4-Manifolds of which there exist no Kirby diagrams

In 4-Manifold theory one makes often the use of Kirby Diagrams to construct 4-manifolds (compact or non-compact) with specific gauge and topological properties (for example small betti numbers, spin ...
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holomorphic exoticness

A topological manifold is an exotic copy of another smooth manifold if it is homeomorphic to it, but not diffeomorphic (and when you switch diffeomorphic by homotopic, you get a fake copy, following ...
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non-orientable 4-manifolds

Most of the books and texts I read about classfication problems surrounding 4-manifolds which are closed and orientable (with a occasional side-track to open orientable 4-manifolds). This is ...
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Looking books about the topology of n-manifold ($n > 4$)

There are a lot of books dealing with the strangeness of the topology of 4-dimensional topology. I wonder if there are books or overview references on the topology of n-manifolds (where n > 4) ? ...
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562 views

Bundle orientability vs manifold orientability

Given a vector bundle, I am a bit hazy about the difference between the notions of its orientability as a bundle and as a manifold. I think I know that the following are true, A tangent bundle of a ...
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152 views

Exotic Manifolds from the inside

As we know, an exotic $\mathbb{R}^4$ is a manifold which is homeomorphic, but not diffeomorphic to the standard $(\mathbb{R}^4,id)$, and there are even very explicit descriptions of them (Kirby ...
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is any subset of a manifold a submanifold?

by definition a submanifold is a subset of a manifold which is itself a manifold. consider $A$ a subset of an $n$-manifold $M$. a neighborhood of $x\in A$ is $\mathbb R^n$ since $x$ is an element of ...
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133 views

Is 2x (360°) rotated (orientable) Möbius band homeomorphic to $S^1\times[0,1]$?

I'm thinking yes, because they are both a quotient of the square. But I can't figure out what the actual homeomorphism is. Do we have to "go outside of $\mathbb{R}^3$" with the homeomorphism? ...
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461 views

Gradient nonzero extensions of a vector field on the circle

Let $\mathbf{v}=(a,b)$ be a smooth vector field on the unit circle $\mathbb{S}^{1}$ such that $a^{2}+b^{2}\neq0$ everywhere in $\mathbb{S}^{1}$ with degree $\deg\mathbf{v}=0$. Suppose also that ...
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1answer
254 views

Sphere eversion and Smale-Hirsch theorem

For two manifolds $M^m$ and $N^n$ with $m<n$ the Smale-Hirsch theorem says that the differential map $d:\operatorname{Imm}(M,N)\to\operatorname{Mon}(TM,TN)$ is a weak homotopy equivalence, where ...
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246 views

Fiber of jet bundle of a fiber bundle

Given a fiber bundle $p:E\to B$ with fiber $V$ and structure group $G$, one can define the corresponding $k$-jet bundle $E^k\subset J^k(B,E)$ of jets of local sections of $E$. On Wikipedia there is a ...
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a neighborhood of an intersection point

if a point $x$ is in the intersection of two spaces $X$ and $Y$ suppose we know explicitly a neighborhood of $x$ in $X$, can we take the same neighborhood of $x$ in $Y$. More specifically, if the ...
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Relationship between the zeros of a vector field and the fixed points of its flow

I'm having a little trouble here and would appreciate some hints. Let $M$ be a compact manifold without boundary and let $X$ be a smooth vector field on $M$ with only isolated zeros. Let $\theta_t$ ...
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1answer
115 views

The prime decomposition of 3-manifold

Let $Y^3$ be a closed 3-manifold with $\pi_1(Y)=\mathbb{Z}$. Is it true that $Y$ is homeomorphic to $S^1\times S^2$ from the prime decomposition of 3-manifold?
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Consequences of Degree Theory

I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable ...
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204 views

If $S^{2n+1}$ is covering space of $X$, then $X$ is orientable.

Is there any direct way to prove that $n$-manifold is orientable? In AT we can just calculate $n$'th homology group and check whether it's $\mathbb Z$ or $0$. But I want a geometric method, using ...
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Which manifolds are parallelizable?

Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
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concrete examples of tangent bundles of smooth manifolds for standard spaces

I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that $$\dim(TM)=2\dim(M).$$ Do the tangent bundles of the following ...
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525 views

Is $M=\{(x,|x|): x \in (-1, 1)\}$ not a differentiable manifold?

Let $M=\{(x,|x|): x \in (-1, 1)\}$. Then there is an atlas with only one coordinate chart $(M, (x, |x|) \mapsto x)$ for $M$. We don't need any coordinate transformation maps to worry about ...
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fundamental group of closed surfaces as CW complexes

Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively. 1) how can I prove, that $nT$ and $nP$ are homeomorphic to ...
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247 views

The Hairy Ball theorem and (non-orientable) real projective plane

Is it possible to prove the Hairy Ball theorem via non-orientability of $P^2(\mathbb{R})$? That is, the non-vanishing section $s \colon S^2 \to TS^2$ would induce (via “2-to-1” bundle $p \colon S^2 ...
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1answer
344 views

Immersive Injections whose images are Embedded Submanifolds

Let $M,N$ be smooth manifolds where the dimension of $M$ is less than or equal to the dimension of $N$. Suppose that $F: M \rightarrow N$ is an injective immersion and $F(M)$ is an embedded ...
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824 views

Why maximal atlas

This has been on the back of my mind for one whole semester now. Its possible that in my stupidity I am missing out on something simple. But, here goes: Let $M$ be a topological manifold. Now, even ...
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precise official definition of a cell complex and CW-complex

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and ...
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No hypersurface with odd Euler characteristic

Here is a classic problem which I encountered and could not solve: Prove that a simply connected closed smooth manifold has no closed sub-manifold of co-dimension $1$ with odd Euler ...
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$f:\mathbb{S}^1\rightarrow\mathbb{S}^1$ odd $\Rightarrow$ $\mathrm{deg}(f)$ odd (Borsuk-Ulam theorem)

I'm having trouble understanding the proof of Borsuk-Ulam theorem ($n=2$) that we did in our class. The only problematic part is the last sentence in the proof of lemma 1. ...
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embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$

Consider the classic map $$F:\mathbb{RP}^2\rightarrow \mathbb{R}^4$$ defined by $$F[x,y,z]=(x^2-y^2,xy,xz,yz)$$. This defines a smooth embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$. It is clearly ...
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normal bundle of level set

Let $M$ be a Riemannian manifold and $S \subset M$ a regular level set of a smooth function $f:M\rightarrow \mathbb{R}^k$. How can I show that the normal bundle of $S$ is trivial? If $k=1$ then ...
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1answer
188 views

Understanding Hom-bundle on vector bundles (Atiyah)

I am trying to make sense of the following statment (from Atiyah's K-theory book) Suppose $V,W$ are vector spaces, and that $E=X \times V, F=X \times W$ are the corresponding product bundles. Then ...
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Compact submanifolds of $\mathbb{R}^n$ without boundary

I'm having a little trouble seeing how to do Exercise 7.5 in Lee Smooth Manifolds: Let $M$ be a smooth compact manifold. Show there is no submersion $F:M\rightarrow\mathbb{R}^k$ for any $k>0$. ...