1
vote
0answers
40 views

Homotopy groups of unitary groups

in this paper I found some explicit generators of homotopy groups of unitary groups, for example $\pi_3[SU_2]$: $\begin{bmatrix}z_1\\z_2\end{bmatrix}$$\rightarrow $$\begin{bmatrix}z_1 ...
6
votes
1answer
138 views

Mathematical background for TQFT

I am physicist. I`ve started studying Topological QFT. What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? What books/articles could help form ...
0
votes
0answers
31 views

Second Volume of Elements of Homotopy Theory?

In the preface of Elements of Homotopy Theory (GTM 61) by George W. Whitehead, he wrote that "I plan to devote a second volume to these developments". Does any one know if George eventually published ...
4
votes
0answers
72 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
40 views

Good resource for learning braid theory?

I recently heard about braid theory and read the Wikipedia article on it, and it seems really beautiful. What is a good resource for learning more about it? I have a background in mathematics at the ...
8
votes
0answers
152 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 ...
2
votes
0answers
76 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 ...
2
votes
4answers
262 views

Text similar to chapter 9 of Topology from James Munkres

I'm self-studying chapter 9 of Topology from James Munkres. I like to read different books about the same topic at the same time. Can someone recommend some text/book that is about the same subjects ...
3
votes
1answer
66 views

Differential equations books using lots of algebraic topology?

The wikipedia page on 'Algebraic Topology' contains the following sentence: One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to ...
2
votes
0answers
76 views

References for Local Orientations and Fundamental Class

I am looking for references with examples about computing induced orientation given a self homeomorphism of a closed orientable n-topological manifold. I have in mind only A.Hatchers Algebraic ...
5
votes
1answer
101 views

Suggestion about Algebraic Topology talk

following the content of the title I am writing here to ask some suggestions concerning a talk I will be presenting at my university in a week or two. The main topic I chose is the fundamental ...
1
vote
0answers
27 views

Explain the terms k-simplex and simplical complex geometrically?

I m new to algebraic topology .so confused with these terms pls suggest simple books
5
votes
0answers
180 views

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
11
votes
4answers
276 views

Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true? Suppose ...
6
votes
0answers
69 views

A theorem of Kan regarding fibrant replacement

Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a ...
0
votes
0answers
21 views

branched cover along a closed curve in the $3$-sphere

Let $c$ be a closed embedded smooth curve in the $3$-sphere $\mathbb S^3$. I was told that $\mathbb S^3$ admits a two fold branched cover $X(c)$, branched along $c$, which corresponds to the ...
1
vote
1answer
40 views

Poincare lemma on current

The Poincare lemma on current states that: If $U$ is a star-shaped open set in $\mathbb R^n$ and $T$ is a $k$-current on $U$ such that $dT=0$, then there is a $k-1$-current $S$ on $U$ such that $dS = ...
6
votes
1answer
91 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
1
vote
1answer
28 views

Connectivity and Euler characteristic for surfaces

I learn the concept of connectivity from Hilbert's Geometry and the Imagination as follows: A polyhedron is said to have connectivity $h$ (or to be $h$-tuply connected) if $h-1$, but not $h$, ...
2
votes
0answers
38 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 ...
1
vote
0answers
75 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
6
votes
2answers
144 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 ...
3
votes
4answers
180 views

What to read alongside with Hatcher Algebraic Topology

I know there are a few posts asking for references about algebraic topology textbooks. Still I have decided to open another one as I would like to ask a slightly different question: which textbook ...
3
votes
0answers
33 views

Is a sufficiently nice simple curve which is nulhomotopic the boundary of a surface?

This is a follow up to Is a simple curve which is nulhomotopic the boundary of a surface?. There, I asked whether, given a simple curve $C$ in an open subset $U$ of $\mathbb R^3$ which is nulhomotopic ...
1
vote
1answer
37 views

Is a simple curve which is nulhomotopic the boundary of a surface?

Let $C$ be a simple curve in an open subset $U$ of $\mathbb R^3$. Suppose that $C$ is nulhomotopic in $U$. Must there exist a homeomorphism $f$ from the closed unit disk $D$ in $\mathbb R^2$ to $U$ ...
1
vote
1answer
65 views

Lefschetz Hyperplane Theorem: reference request

I've just begun working on my bachelor thesis on the "Lefschetz Theorem on Hyperplane Sections" (see for example http://en.wikipedia.org/wiki/Lefschetz_hyperplane_theorem). The goal of the thesis is ...
1
vote
2answers
61 views

Reference request for bounded cohomology

I want to read Gromov's IHES paper Volume and bounded cohomolgy. I have a decent background in algebraic topology at the level of Hatcher. What other background is required to understand the landmark ...
1
vote
1answer
86 views

What is a 3D winding number?

This paper mentions the term "3D winding number". Its abstract says: "We develop a new formulation, mathematically elegant, to detect critical points of 3D scalar images. It is based on a topological ...
2
votes
2answers
84 views

Self-contained text on characteristic classes

I am looking for a clear, self-contained text (either a book or lecture notes) that deals with characteristic classes, starting from the very basics (fiber bundle, principal bundle etc.), and ...
5
votes
2answers
131 views

Good source for a point set topological introduction to CW complexes?

Most algebraic topology books I found don't dwell too much on point set topology of CW complexes. I'd like too become more familiar with them. Anyone knows a good source (with exercises) too learn ...
1
vote
0answers
14 views

Looking for a basic reference on propagators (in Topology)

I am looking for a basic (preferably self-contained) reference where I can read about propagators (as they appear in Topology), and in particular Morse propagators. Thanks!
14
votes
7answers
470 views

Introduction to ring theory?

I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear ...
10
votes
1answer
299 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 ...
4
votes
1answer
101 views

Contracting a contractible set in $\mathbb R^2$

Assume that $A$ is compact, connected and contractible set in $\mathbb{R}^{2}$ (for example: simple square). If we contract this set to a point the space still will be homeomorphic to ...
0
votes
0answers
38 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 ...
7
votes
5answers
297 views

Undergraduate-level intro to homotopy

I'm looking for an undergraduate-level introduction to homotopy theory. I'd prefer a brief (<200pp.) book devoted solely/primarily to this topic. IOW, something in the spirit of the AMS Student ...
5
votes
1answer
127 views

Prerequisites for bredon's “Topology and Geometry”?

My background in topology is the first 6 chapters of Munkres's "topology" and in algebra Herstein's "topics in algebra". Both of them I self studied. A look at the table of contents of bredon's ...
3
votes
2answers
186 views

Hopf fibration and $\pi_3(\mathbb{S}^2)$

I am interested in Hopf's original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre ...
4
votes
1answer
106 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 ...
4
votes
0answers
107 views

Relationship between paradoxes in logic and geometry/topology

Though I've been reading for years, this is my first question here. Believe it or not, I've tried the search feature- apologies if this is a duplicate. The main point of this post can be summarized ...
0
votes
0answers
63 views

Ham Sandwich theorem used in combinatorics problem involving beads on a necklace

Ok, so according to a friend of mine you can use the ham sandwich to prove the following theorem: Suppose there is a necklace with $m$ types of beads and $2n_1,2n_2...2n_m$ beads of each colors. So ...
1
vote
0answers
58 views

Different versions of Hatcher

I suddenly found out that my Hatcher from amazon is very different from the version on his website. Should I assume his website is up to date, and hence my copy is an old version?
10
votes
1answer
111 views

Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
6
votes
1answer
398 views

Long exact sequence for cohomology with compact supports

Related to my previous question here. Let $X$ be a topological space and let $H_c^{\bullet}(X)$ denote its singular cohomology with compact supports (rational coefficients). Let $U$ be an open subset ...
2
votes
2answers
105 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
1answer
82 views

If compact simply connected manifold has the same rational homotopy groups as $S^n$ or $\mathbb{C}P^n$, must it have the same cohomology ring?

The question came up while trying to shorten a paper I'm writing into submission-ready length. Let $M$ be a compact simply connected manifold. By defininition, the rational homotopy groups of $M$ ...
0
votes
0answers
29 views

triangulated polyhedron

I want to improve my knowledge about Topology. I am going to start the topic "triangulated polyhedron". But i have not know anything about it at this moment. I need a perfect reference that helps me ...
5
votes
3answers
319 views

References for Topology with applications in Engineering, Computer Science, Robotics

I am reading a book on motion planning for mobile robots, I have a really hard time with the mathematics. Some parts it is talking about the topology of the space and manifolds and compactness of the ...
5
votes
2answers
207 views

Who proved that existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ was sufficient for HEP?

It is well known that the existence of a retraction $r:X\times\mathbb{I}\rightarrow X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is necessary to make $\left(X,A\right)$ a pair having the homotopy ...
4
votes
1answer
198 views

constructing a genus 2 surface from 8-gon

I am requesting some help or reference for visualization? I am having a hard time constructing a genus 2 surface from 8-gon. May I request for some reference? Here's the construction I used from ...