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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Prove that it is a regular point

Good night, my friends, can you help me with these exercise? Let $N$ a $k$-manifold, $X$ a compact $(k+1)$-manifold in $\mathbb R^N$, and $F:X\rightarrow N$ a differentiable map. Let $y\in ...
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Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
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118 views

Normal bundle of the two-dimensional sphere manifold embedded in $\mathbb R^4$

Let $M \subset \mathbb R^4$ be a smooth manifold diffeomorphic to $S^2$. How can one prove that normal bundle of $M$ has at least one non-vanishing global section. I think that $M$ should be ...
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40 views

Euler class is odd under orientation, thus its integral over a manifold will be even.

I learned a statement from others: "Euler class is odd under orientation, thus its integral over a manifold $M$ is even." I cannot fully appreciate it, can someone show this explicitly? The ...
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1answer
333 views

two interlocked circles are homeomorphic to two noninterlocked circles

This is what I learned from here the post: two interlocked circles are homeomorphic to two noninterlocked circles, thus they (two interlocked circles and two noninterlocked circles) are homotopic ...
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51 views

Avoiding self-intersections of immersed manifolds

Let $i: N\to M$ be an immersion of manifolds. If $\dim M\geq 2\dim N+1$ (or something like that?), does there exist arbitrary small perturbations of $i$ (wrt. some reasonable norm) that are already ...
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201 views

Fixed point theorem on spheres

In Milnor's book Topology from the Differentiable Viewpoint there's the following problem: Problem $6$ (Brouwer). Show that any map $S^n\to S^n$ with degree different from $(-1)^{n+1}$ must have a ...
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163 views

proving jordan brouwer separation theorem

For proving Jordan separation theorem in differential manifold theory, one step involves proving the following: Let $z\in{\mathbb{R}^n}\setminus X$, where $X$ is connected, closed manifold of dimensin ...
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1k views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
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56 views

A basic question on intuition of second countability for definition of topological manifold

What is the intuition behind the requirement of second countability in definition of topological manifold? It seems that it is relevant to the $\mathbb{R}^n$ has second countable basis. Is there any ...
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1answer
49 views

invariants of a Lie algebra

What does it mean by "constructing invariants" in algebraic topology or algebra in general? How to define a "invariant" in algebra? What does it mean by the "invariant of a Lie algebra"?
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174 views

Which manifolds are zero sets of $\mathbb R^n$ valued maps

If $M$ is a smooth manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal bundle induced by $f$. ...
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226 views

Euler characteristic for non-compact manifolds

How can one generalize the Euler characteristic to non-compact manifolds? Furthermore, is there a way to generalize the notion of an intersection number to non-compact manifolds, so that one could ...
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1answer
146 views

Another differential topology lemma

Another lemma (1) Why can we assume $z=f(z)=0$ and that $U$ is convex? (the coordinate domains of the manifolds can be taken to be balls?) (2) Why is it enough to consider the special case of a ...
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56 views

Set of diffeomorphisms on a manifold

It is well known that given a compact smooth boundaryless manifold $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r \geq 1$, is open in $C^{r}(M)$, the set of continuous functions (for ...
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106 views

Why surgery produce a new 3-manifold?

I was studying a proof of the fact that any closed orientable 3-manifold is obtained by integer surgery along a link. I read the several proofs but I don't understand well. A proof is as follows. ...
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1answer
171 views

A differential topology lemma

Consider the following lemma (1) How come he talks about degrees here, after all he doesn't assume $X$ to be oriented? (2) Why is $\bar{v}|\partial X$ homotopic to $g$? (NOTE: we consider them as ...
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47 views

the second projection is a local diffeomorphism between manifods with boundary

Good afternoon, I Have problems with these exercises, I wrote the two questions because the second one needs the result of the previus one. They are: Let $n,m, k=n+m$ positive integers, so we can ...
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146 views

Homology of manifolds with boundary

If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is ...
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24 views

Example of a map which is not a diffeomorphism

Can anyone think of a bijective smooth map from a compact space to a huasdorff space which is not a diffeomorphism? thanks
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235 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
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760 views

Simply connected manifolds are orientable

For a simply connected $n$-manifold $M\subseteq\Bbb{R}^k$, I want to show that $M$ is orientable. Take a point $p\in M$ and take an $n$-disc, $D^n$, around $p$ (we can take it as small as we please). ...
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46 views

A consequence of simply connectedness

Is it true that if a manifold is simply connected then its tangent bundle is simply connected?
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42 views

How to calculate the Sum of a differential map?

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ a differentiable map such that $f(x) = x$ for $x \in [-T,T]$, $T>0$ and such that $0$ is a regular value. Calculate ${\sum}_{x\in f^{-1}(0)} ...
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45 views

Why is this statement about integral curves correct?

The following definitions and example are taken from John Lee's Smooth Manifolds, 2nd edition. Given a vector field $V$ on a smooth manifold $M$, we define an integral curve of $V$ to be a ...
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226 views

De Rham cohomology, and forms on manifolds

In String Theory and M-Theory by Becker, Becker and Schwarz, they introduce a group, $$C^{p}(M)$$ which they denote the group of all closed $p$-forms on the manifold $M$. Furthermore, they state ...
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When $f\colon M\to N\times N$ satisfies that $f^{-1}(\Delta)$ is a ball?

Let $M$, $N$ be smooth manifolds of dimension $m+n$ and $n$, respectively. Suppose that $f\colon M\to N\times N$ is a smooth map and $f$ is transversal to the diagonal $\Delta=\{(x,x)\in N\times ...
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126 views

Reviewing the basics of algebraic topology for further deeper study

I am entering a Ph.D. program in pure math this fall. Over the summer, I am hoping to review the basics of algebraic topology (fundamental group, covering space theory, etc.) and I wondered if you had ...
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127 views

Cauchy's Theorem by Differential Geometry

Is there a prove of Cauchy's theorem footing on the topology of the complex plane (homotopy, differential forms, etc.)? More specific consider a differentiable Banach space valued complex function. ...
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35 views

Making foliations C^1 smooth

I am not an expert in foliations, but I'm interested in the following conjecture of Cantwell and Conlon (from "An interesting class of $C^1$ foliations"): "If $F$ is a transversely orientable ...
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1answer
163 views

Is there any embedding theorem for fibre bundles?

I would like to know whether there is an embedding theorem for fibre bundles, like Whitney embedding theorem. When can a given fibre bundle be a subbundle of some higher dimensional bundle?
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182 views

smooth surjective map from lower dimensional manifold onto higher dimensional manifold?

I'm thinking about whether there exists a smooth surjective map $f: M^m \twoheadrightarrow N^n$ where both $M$ and $N$ are smooth manifolds, and $\dim M = m < n = \dim N$. (We might assume that $M$ ...
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express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
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show that $\omega$ is exact if and only if the integral of $\omega$ over every p-cycle is 0

Let $M$ be an oriented smooth manifold and $\omega$ a closed $p$-form on $M$. Show that $\omega$ is exact if and only if the integral of $\omega$ over every $p$-cycle is $0$. In particular, how to ...
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276 views

show that the compactly supported De Rham cohomology groups $H^{p}_{DR}(\mathbb{R^n})$ are all zero for $0\leq p<n$

I am reading John Lee's book and there is a problem on De Rham cohomology: show that the compactly supported De Rham cohomology groups $H^{p}_{DR}(\mathbb{R^n})$ are all zero for $0\leq p<n$. My ...
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1answer
25 views

generation of sub bundle

Let $M$ differentiable manifold with $\dim M=n$. If $(TM,\pi,M)$ be the fiber bundle tangent. Consider the family $E=\lbrace E_x\rbrace _{x\in M}$ such that $E_x \subset T_xM$ and $\dim E_x=k$ for ...
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50 views

Finding zeros of maps between manifolds with different dimensions

One of the questions for which the notion of degree is useful is: does this map have a zero. For example, one can prove the Fundamental Theorem of Algebra using the following fact involving the ...
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Are $C^{k}$ manifolds the same as $C^{\infty}$ manifolds?

This is a theorem of Hassler Whitney: For $0<k<\infty$ and any $n$-dimensional $C^k$ manifold the maximal atlas contains a $C^\infty$ atlas on the same underlying set. It seems to me ...
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63 views

Constructing maps of degree $k$

One of the common constructions one finds when first learning about the (topological) degree of a map is the construction of maps $f_k:S^n\rightarrow S^n$ of degree $k\in\mathbb{Z}$ (i.e. ...
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31 views

About embeddings of connected sums

Let $M_1$ and $M_2$ be two soomth manifolds who're already embedded in $\mathbf{R}^k$. Can one prove that the connected sum of $M_1$ and $M_2$ can also be embedded into $\mathbf{R}^k$ ?
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189 views

Transverse intersection in a compact manifold

Is it true that if $M$ is a compact manifold and $X,Y$ are submanifolds of $M$ which intersect transversely that the intersection $X\cap Y$ consists of finitely many points? I'm trying to understand ...
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1answer
245 views

Is the Hopf fibration the only fibration with total space $S^{3}$?

Let $S^{1}$ bundle over $S^{2}$ is homeomorphic (diffeomorphic) to $S^{3}$. Is the Chern class of the fibration 1, i.e. is the Hopf fibration up to isomorphism the only fibre bundle with total space ...
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121 views

the knot surgery - from a $6^3_2$ knot to a $3_1$ trefoil knot

It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot: e.g. from to All one needs to do it to cut the three intersections at the angle of ...
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let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
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216 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
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Three linearly independent vector fields

How can one find three linearly independent vector fields on $S^1\times S^2$? I know that $S^1\times S^2 \cong SO_3( \mathbb{R})$, i.e. the set of orthogonal $3 \times 3$ matrices with determinant ...
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70 views

Structure of level sets of a noncritical point of a smooth function on a two dimensional domain

Let $\psi$ be a smooth function on a two dimensional simply connected domain $\Omega$ such that $\psi=0$ on the boundary $\partial \Omega$. Suppose $\rho$ is not a critical value of $\psi$ then it is ...
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$U(n)$ and $U(1)\times SU(n)$ are not isomorphic Lie groups

I am reading John Lee and on the chapter about group actions there is a problem that asks me to show that $U(n)$ and $U(1)\times SU(n)$ are not isomorphic Lie groups by showing that they don't have ...
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Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
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112 views

Orientation of manifolds with boundary

I have an ambiguity about how to orient the boundary of a manifold. In particular : Consider the example $M=B^2 \subset \mathbb{R^2}$ be the manifold with boundary. suppose positive orientation for M ...