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.

learn more… | top users | synonyms

8
votes
2answers
223 views

Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
7
votes
0answers
86 views

Definition of bordism - gluing manifolds with structure

In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U ...
1
vote
1answer
35 views

Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
4
votes
0answers
75 views

Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
2
votes
1answer
49 views

Intuition behind the definition of a derivative by Lang

In Serge Lang's Introduction to Differentiable Manifolds he says that a function $f:U\to F$ is differentiable at a point $x_0\in U$ if there exists a linear map $\lambda$ of $E$ into $F$ such that, if ...
2
votes
3answers
91 views

What is the pushforward of a function (not a vector)

If we have two manifolds $M$, $N$ with the map $f:M \to N$, then this induces a map between their tangent spaces $f_*:T_pM \to T_{f(p)} N$. By duality, another map exists $f^* : T^*_{f(p)}N \to ...
0
votes
0answers
20 views

Question on framed bordism classes definition

I was reading recently about cobordism, and in specific about the Thom-Pontraygin theorem which states $\pi_{k}(S^n)$ is isomorphic to the cobordism classes of framed $n$-manifolds in $R^k$. In ...
0
votes
0answers
39 views

Reference requestion: Existence/construction of bump functions

I'm not much of an analyst myself, but I've time and time again come across proofs which require knowledge of the existence of bump functions. However, I've never studied them, so I'm missing ...
10
votes
1answer
225 views

Am I reading Bott - Tu right?

Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do ...
0
votes
1answer
33 views

homotopying a function to itself by a regular homotopy

I wonder whether there is a smooth function $f:\mathbb{S}^{1}\rightarrow \mathbb{R}^{1}$ which may be homotoped to itself by a regular homotopy $H(x,t)$, i.e. by a smooth ...
2
votes
0answers
61 views

Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
5
votes
1answer
113 views

Poincare dual of unit circle

I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in $ R^2-\{0 \} $ is zero, ...
1
vote
1answer
93 views

Change of Coordinate Formula for Differential Forms

Let $M$ be a manifold, $x$ local coordinates on an open set $U$, $y$ local coordinates on an open set $V$. In addition, let $(x, \alpha)$ and $(y, \beta)$ be two induced bases for the common part of ...
0
votes
1answer
36 views

Linking number of a pair of circles

Definition Given disjoint manifolds $M$, $N\subset \Bbb{R}^{k+1}$, the linking map $\lambda: M\times N\to S^k$ is defined by $\lambda(p, q) = (p - q)/||p - q||$. If $M$ and $N$ are compact, oriented, ...
0
votes
0answers
52 views

Turning a torus inside-out

Smale's paradox is now famous, and great videos can be found illustrating it. Similarly, there is a video showing how to turn a torus inside-out. The solution seems to be simpler, but is the proof ...
4
votes
1answer
151 views

Metric Tensors and its Taylor Expansion in Normal Coordinates

With metric tensors of the unit sphere in normal coordinates, their Taylor series for $p\in S$ near the north pole $N$ can be written as follows. $$g_{rr}(p) \equiv 1; g_{r\theta}(p) = g_{\theta ...
1
vote
1answer
76 views

Reference about Gauss-Bonnet-Chern theorem.

I would like to get some references which explains Gauss-Bonnet-Chern theorem and its original proof by Chern. I tried to read his paper published in 1944 "A Simple Intrinsic Proof of the Gauss-Bonnet ...
3
votes
1answer
80 views

Metric Tensors in Geodesic Normal Coordinates

Consider the unit sphere and its north pole $(0, 0, 1)$. My question is how to write the metric tensor $g_{ij}$ in geodesic normal coordinates. I know that metric tensors are defined as inner product ...
3
votes
1answer
78 views

Looking for a good book on Morse-Bott functions.

I am looking for a book to study for the first time Morse-Bott functions. Does anyone know one that is easy to follow and detailed? If there is one connecting this subject with symplectic geometry, it ...
1
vote
1answer
30 views

DeRham Chohomology of the Circle and the Torus

I want to compute the first DeRham Chohomologygroup of the circle, thus in symbols $H^1_{dR}(S^1)$. Let $p(x)=e^{ix}$ the map from $\mathbb{R}$ to the circle $S^1$, $\Omega^1(S^1)$ the set of all ...
1
vote
1answer
30 views

differentiable map on sphere

I'm trying to show that if $f:S^n\to \mathbb{R}$ is differentiable, then there are two distinct points $p,q\in S^n$ where the differentials $T_pf$ and $T_qf$ vanish. Any suggestions?
0
votes
0answers
32 views

Neighbourhood of set of functions!

I am going through some topology papers and i encountered strange notation of neighbourhood (of set of functions) which I don't understand. Can you please elaborate that notation for me? Here it ...
5
votes
0answers
44 views

How to kill homotopy groups using framed cobordism

Let $M$ be an orientable manifold (with or without boundary), $N$ a framed submanifold in the interior of $M$ and assume (if necessary) that $\dim N<(\dim M)/2$. If some low-dimensional homotopy ...
2
votes
1answer
38 views

Wedge product of basis elements of cohomology

Let $M$ be a compact, connected, oriented 4-manifold without boundary. If $H^2(M)\cong \mathbb{R}^2$ and I have a basis $\{[\omega_1],[\omega_2]\}$ for $H^2(M)$, is it the case that $[\omega_1\wedge ...
4
votes
0answers
67 views

A question on harmonic two-forms

Let $(M^4,g)$ be a closed Riemannian four-manifold with $b_2^+>0$ and $b_2^->0$, is it possible to find two harmonic two-forms $\alpha\in H^2_+(M)$ and $\beta\in H^2_-(M)$, such that ...
0
votes
1answer
51 views

When is an exact 2-form harmonic?

Let $\alpha$ be an exact two-form, $\alpha=d\beta$ for some one-form $\beta$, when is $\alpha$ harmonic? By uniqueness of harmonic forms in cohomology classes, it cannot be harmonic?
1
vote
0answers
71 views

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 ...
0
votes
0answers
46 views

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 ...
3
votes
1answer
57 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 ...
0
votes
1answer
26 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 ...
1
vote
1answer
161 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 ...
1
vote
1answer
25 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 ...
3
votes
1answer
50 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 ...
2
votes
0answers
44 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 ...
1
vote
2answers
173 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 ...
1
vote
0answers
44 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 ...
0
votes
1answer
36 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"?
6
votes
0answers
139 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 $f$. ...
2
votes
1answer
68 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 ...
1
vote
1answer
136 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 ...
1
vote
0answers
47 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 ...
0
votes
0answers
72 views

Two books defined two Chern(Euler) classes yet differed by a negative sign, what's wrong?

In the book 'Principles of Algebraic Geometry' P141 and the book 'Differential Forms in Algebraic Topology' P72-73, they defined the Chern(Euler) classes of line bundles using the patching data ...
0
votes
0answers
52 views

Flow of Linear Vector Fields

The following is a statement from "Notes on the Topology of Vector Fields and Flows" by Daniel Asimov. In the case where a vector field on $\mathbb R^n$ is defined by a matrix, then there is a simple ...
3
votes
1answer
75 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. ...
0
votes
1answer
167 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 ...
1
vote
0answers
33 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 ...
1
vote
2answers
95 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 ...
0
votes
2answers
19 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
1
vote
0answers
81 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 ...
2
votes
3answers
125 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). ...