0
votes
0answers
14 views

Subsets of immersions that are embeddings

Let $X$ be a manifold and let $Y$ be a submanifold, possibly with boundary. I am dealing with a situation where $f:X\to \mathbb{R^3}$ is an immersion, but $f\vert_Y:Y\to \mathbb{R}^3$ is an embedding. ...
2
votes
0answers
48 views

handle moves: proof

In several 4-manifold textbooks, when handle moves (creation, cancellation, sliding) are discussed, they are explained using very helpful drawings. However, I would like to know if there is a ...
2
votes
0answers
29 views

Reference about history of characteristic classes

I'm looking for a good reference about history of characteristic classes. For example, I would like to know how Chern define the Chern class at first in history, or who rewrote the characteristic ...
6
votes
2answers
124 views

Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such ...
2
votes
0answers
48 views

The first proof for Poincare lemma in history

How can I get a reference about the first proof of Poincare lemma in history? I already know some methods of proof, but I do want to know the original approach. Thanks for your help!
9
votes
1answer
191 views

Preparing for “differential forms in algebraic topology”?

I'd very much like to read "differential forms in algebraic topology". Apart from background in calculus and linear algbra I've thoroughly went through the first 5 chapters of Munkres. I'm thinking ...
0
votes
0answers
37 views

Some questions on definition of Grassman manifolds

In W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry , Page 64. We use coordinate correspondences $\phi_j : U_j \to R^{k(n-k)}$ in order to define Grassman manifolds ...
0
votes
1answer
132 views

Reference request : Study of Differential topology post Milnor's book

I am just about to finish my study of Milnor's book 'Topology from the Differentiable Viewpoint' and I really love the subject. I would like to continue my study of Differential Topology and am ...
2
votes
0answers
38 views

Definition of linking number for disjoint submanifolds of the sphere- A problem from Milnor's book

In problem number 13 of Milnor's 'Topology from the Differentiable Viewpoint', the linking number for two compact boundary-less manifolds $M,N \subset \mathbb{R}^{k+1}$ of dimensions $m,n$ such that ...
0
votes
1answer
46 views

Morse Sard Theorem

I'm reading the Morse-Sard's theorem in the book Differential Topology by Hirsch, and I wonder if anyone has a paper on the theorem with more details? I'm not proving it for manifolds, just open sets ...
1
vote
0answers
33 views

Reference: forms invariant under Lie group action give the de Rham cohomology?

I'm looking for a reference for a proof of the following fact: Let $G$ be a compact, connected Lie group acting on a smooth manifold $M$. Then inclusion of the differential forms invariant under the ...
3
votes
1answer
77 views

Hopf invariant and the linking number

The Hopf invariant of a map $f:S^{2n-1}\to S^n$ can be defined in various ways, in particular: (1) as the linking number of the preimages of two points and (2) using the cohomology ring of the space ...
17
votes
1answer
121 views

Reconstructing a manifold from critical points

I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked ...
1
vote
1answer
119 views

Why are normal bundles always locally trivial?

Is there a quick and dirty proof that normal bundles (say of some submanifold in a smooth manifold) are always locally trivial? My notes seem to have swept this assumption under the rug. Even ...
2
votes
1answer
80 views

Compact manifolds can almost be immersed?

The Whitney Immersion theorem states that any $n$-dimensional manifold can be immersed in $\mathbb{R}^{2n}$. However, I seem to remember that if $X$ is a compact $n$-dim manifold, then $X$ can be ...
2
votes
2answers
98 views

Extending diffeomorphism to disk

I am trying to prove the following If $f:S^1 \to S^1$ is a diffeomorphism it can be extended to a diffeomorphism $F: D^2 \to D^2$. But I can't seem to prove it. I proved it for homeomorphisms using ...
5
votes
0answers
51 views

The space of paths

Let $ (M,g) $ be a compact Riemannian manifold, and let the path space $ \Omega $ of $ M $ be the set of equivalence class of smooth maps $ \gamma : [0,1] \rightarrow M $ (equivalent under ...
6
votes
1answer
175 views

Isotopy preserving inverse image $f_t^{-1}(V)$ of a homotopy

During a lecture I was given a bunch of easy propositions entitled as "observations" by the lecturer. But one of them seems more difficult and I have absolutely no idea how to "observe" it... I'd be ...
1
vote
1answer
160 views

Thinking of giving up..

I got really stuck to the end of Guillemin and Pollack (in particular, here) and plan to give up. Give up Guillemin and Pollack, not math though. It seems John Milnor's classic little book topology ...
4
votes
2answers
130 views

Textbooks with exposition done mostly in proof outlines or exercises?

As the title indicates, I'm trying to find books where the exposition of the main course of thought is done entirely or mostly in outlines of proofs, or as exercises with or without hints. I'm trying ...
0
votes
2answers
218 views

Euler characteristic of an $n$-sphere is $1 + (-1)^n$.

I am using the textbook Guillemin and Pollack's Differential Topology but I am asked to solve a question need this fact that Euler characteristic of sphere is $1 + (-1)^n$. So may I ask if this is ...
3
votes
1answer
163 views

Differential topology book

I want to self-study differential topology. I'd like to hear suggestions from you about appropriate books that I could use while studying. Note: I have not studied differential topology before. I ...
2
votes
1answer
52 views

Quantitative Transversality

I came across the following problem while reading some literature in Dynamical Systems. Say I have an ambient Riemannian manifold $(M,g)$ and a pair of transverse embedded disks $D_1, D_2$ of ...
3
votes
2answers
94 views

How is the norm on $C^k(M)$ defined?

Let $M$ be a smooth, compact $n$-dimensional manifold and $C^k(M)$ the space of real valued $C^k$-maps on $M$. I am looking for a definition of the norm $|\cdot|_k$ on $C^k(M)$ that induces the ...
1
vote
0answers
39 views

Computing group of exotic spheres

In Levine on page 90 it is stated that the following sequence is exact $$ 0 \to bP^{n+1} \to \Theta^n \to Coker(J_n) $$ where $\Theta^n$ is the group of exotic spheres, $bP^{n+1}$ is the subgroup of ...
2
votes
1answer
40 views

The map $S_n \to \Pi_n / J_n$

Let $S_n$ denote the set of all oriented diffeomorphism classes of closed smooth homotopy n-spheres. Let $S_n^{bp}\subseteq S_n$ denote the subgroup represented by homotopy spheres that bound ...
1
vote
2answers
74 views

Groups of homotopy spheres II?

Where can I find "Groups of homotopy spheres: II", the sequel to "Groups of homotopy spheres: I"?
11
votes
0answers
219 views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
3
votes
1answer
59 views

normal form of an n-form

It is known, that one can convert any function $f(x_1,\dots,x_n)$, defined near $0$, into the function $(y_1,\dots,y_n)\mapsto a+y_1$, by a suitable local change of coordinates, provided $df\neq 0$. ...
7
votes
2answers
146 views

Example of a diffeomorphism of class $C^{k}$ which is not $C^{k+1}$

Can anyone give me an example of a map $f:\mathbb{R}\to\mathbb{R}$, which is a diffeomorphism of class $C^{k}$ but it is not a diffeomorphism of class $C^{k+1}$?
1
vote
0answers
118 views

An example of a differentiable manifold class $C^k$ but not class $C^{k +1} $

I'm looking for an example of a differentiable manifold of class $C^k$ but not class $C^{k +1}.$ I found an exercise in Hirsh's book, which suggests that the graph of $f (x) = |x|^{\lambda}$, where ...
2
votes
0answers
90 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
2
votes
0answers
56 views

smoothness structure on a set

I'm reading Milnor's 'characteristic classes', and in the first chapter he defines smoothness structure on a set M, which is confusing for me, as what follows:(these are not Milnor's words so please ...
8
votes
1answer
142 views

Classification of all 28 Exotic 7-Spheres

In http://en.wikipedia.org/wiki/Exotic_sphere#Explicit_examples_of_exotic_spheres Wikipedia says "As shown by Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection ...
1
vote
1answer
416 views

where can I find solutions to A comprehensive introduction to differential geometry by Spivak?

I have tried google and I fail to find solutions to the exercises in the book A comprehensive Introduction to differenial geometry volume I by Spivak. Does anyone know about a site with solutions to ...
2
votes
1answer
258 views

control of the $C^{1}$ norm of a diffeomorphism

Let $\mathcal{E}$ be the set of smooth manifolds with boundary $E\subset \mathbb{R}^{3}$ which are perturbations of the unit ball whose volume $V$, diameter $d$ and area of the boundary $A$ satisfy: ...
2
votes
0answers
107 views

Reference Request: Poincare-Hopf Index Theorem

When I read Griffiths' Algebraic Curves page 23, it states Poincare-Hopf formula for differential forms that is smooth except finite isolated singularities. I tried to find reference for it, but ...
2
votes
0answers
99 views

Are strongly close maps homotopic?

While reading about various results related to density of smooth functions in the space of continuous functions with strong topology, I've got the impression that it is a general fact that for any ...
19
votes
1answer
378 views

How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
5
votes
1answer
134 views

Proving that $O(n,m)$ is simply connected.

My question is the following: Under which conditions on given integers $n\le m$ is $$O(n,m) = \{A \in \mathbb R^{m\times n} : A^TA = \mathbf 1\}$$ simply connected? Does anyone know a reference for ...
0
votes
1answer
229 views

Co-homology Groups of the Torus

I wanted to explain to me, or give me a reference of how to calculate the cohomology groups of the complex and real, torus $\mathbb{T}^2$. I want to use this as an example in a seminar that I will ...
2
votes
1answer
143 views

Reference for topology and fiber bundle

I am looking for an introductory book that explains the relations of topology and bundles. I know a basic topology and algebraic topology. But I don't know much about bundles. I want a book that ...
5
votes
1answer
149 views

On the converse of Sard's theorem

Let $f: M \rightarrow N$ be a smooth map between two submanifolds of $\mathbb{R}^{m}$, $\mathbb{R}^{n}$ respectively. Sard's famous theorem asserts that the set of critical values $C$ of $f$ has ...
3
votes
3answers
330 views

Prerequisites for studying smooth manifold theory?

I am attending first year graduate school in about three weeks and one of the courses I am taking is an introduction to smooth manifolds. Unfortunately, my topology knowledge is minimal, limited to ...
5
votes
1answer
194 views

Any manifold admits a morse function with one minimum and one maximum

I have heard the claim: "Any closed manifold admits a Morse function which has one local minimum and one local maximum" often used in talks without a reference. This does not seem to be very easy to ...
2
votes
0answers
169 views

Hairy ball theorem: references to applications

I'm looking for references to applications of the Hairy ball theorem. I already visited wikipedia and cited references, but I need a little more explanation in both meteorology and applications in ...
2
votes
0answers
143 views

Poincaré-Hopf theorem using Stokes

The wiki entry on the Poincaré-Hopf theorem claims that it "relies heavily on integral, and, in particular, Stokes' theorem". However, in the sketch of proof given there which is more or less the one ...
2
votes
1answer
116 views

Showing $[f^{-1}(y)]$ is Poincare dual to $f^*(\operatorname{vol})$.

Let $f: N^n \to M^m$ be a smooth map between closed oriented manifolds. Then I'm trying to show that for almost all $y \in M$, the homology class $[f^{-1}(y)] \in H_{n-m}(N)$ is Poincare dual to $f^* ...
5
votes
0answers
309 views

Higher-order derivatives in manifolds

If $E, F$ are real finite dimensional vector spaces and $\mu\colon E \to F$, we can speak of a (total) derivative of $\mu$ in Fréchet sense: $D\mu$, if it exists, is the unique mapping from $E$ to ...
4
votes
1answer
114 views

Tubular neighbourhood style theorem reference request

Let $X$ be a smooth manifold and $Y$ be a closed submanifold. Then there exists a neighbourhood $U$ of $Y$ in $X$ such that $Y$ is a deformation retract of $U$ right? I can only find (stronger forms ...