What is Algebraic topology: Definition and 56 Discussions
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
The Math challenge threads have returned!
Rules:
1. You may use google to look for anything except the actual problems themselves (or very close relatives).
2. Do not cite theorems that trivialize the problem you're solving.
3. Do not solve problems that are way below your level. Some problems...
Hi everyone, I'm fibrebundle. I actually joined this forum because I'm really interested in abstract maths. I'm particularly intereseted in alegebraic topology and geometry at the moment. But I'm also really interested in spectral graph and graph theory. I'm starting grad school in engineering...
So before I start I technically do now that the group I am dealing with is just a representation of the Klein bottle but I am not supposed to use that as a fact because the goal of the problem is to derive that information.
Problem:
Let G be a group of with two generators a and b such that aba...
prove that if ##g:Y→Z## and ##f:X→Y## are two smooth maps between a smooth manifolds, then a homomorphism that induced are fulfilling :## (g◦f)∗=f∗◦g∗\, :\, H∙(Z)→H∙(X)##
I must to prove this by a differential forms, but I do not how I can use them .
I began in this way:
if f∗ : H(Y)→H(X), g∗...
in the tv show "The Big Bang Theory", Sheldon wrote a book called "A proof the algebraic topology can never have a non self-contradictory set of abelian groups". Is this just a random set of words that is meant to sound smart but in reality means nothing or is it accurate? If it is, what does it...
I am newbie to topology and trying to understand covering maps and quotient maps. At first sight it seems the two are closely related. For example SO(3) is double covered by SU(2) and is also the quotient SU(2)/ℤ2 so the 2 maps appear to be equivalent. Likewise, for ℝ and S1. However, I...
Eugene Wigner once famously talked about the "unreasonable effectiveness of mathematics" in describing the natural world. Today again we are seeing this in action in particular with regard to the description of the biological brain from the perspective of neuroscience. Researchers from the Blue...
Homework Statement
Prove that any simplicial complex is Hausdorff.
Homework EquationsThe Attempt at a Solution
I have proved that for any finite simplicial complex, it is metrizable and hence Hausdorff.
How to show the statement for infinite case?
I am self leaning some basic cohomology theory and I managed to go through from the definition to the universal coefficient theorem. But I don't think I get the main point of this theory, I like to ask this questions:
Is such an abstract theory practical?
I would say that homology is...
Consider an algebraic variety, X which is a smooth algebraic manifold specified as the zero set of a known polynomial.
I would appreciate resource recommendations preferably or an outline of approaches as to how one can compute the period matrix of X, or more precisely, of the Jacobian variety...
I am a physics student trying to self-learn Chern numbers and Chern class. The book I am learning (Nakahara) introduces the total Chern class as an invariant polynomial of local curvature form ##F##
## P(F) = \det (I + t\frac{{iF}}{{2\pi }}) = \sum\limits_{r = 0}^k {{t^r}{P_r}(F)} ##
and each...
I am with a query about cw complex. I was thinking if is possible exist a cw complex without being of Hausdorff space. Because i was thinking that when you do a cell decomposition of a space (without being of Hausdorff) you do not obtain a 0-cell. If can exist a cw complex with space without...
I am studying topology right now and am a bit confused about the idea of triangulation. The definition is: if a topological space X is homeomorphic to a polyhedron K (union of simplexes) then X is triangulable and K is a (not necessarily unique) triangulation.
Apparently ## K_0 \equiv {1} \cup...
Hello everyone, I'm a undergraduate at UC Berkeley. I'm doing theoretical physics but technically I'm a math major. I really want to study quantum gravity in the future. Now I have a problem of choosing courses. For next semester, I have only one spot available for either representation theory...
i: B to Y is an inclusion, p: X to Y is a covering map. Define $D=p^{-1}(B)$, we assume here B and Y are locally path-connected and semi-locally simply connected. The question 1: if B,Y, X are path-connected in what case D is path-connected (dependent on the fundamental groups)? 2 What's the...
I was watching this video on Abstract Algebra and the professor was discussing how at one point a few mathematicians conjectured the special orthogonal group in ##\mathbb{R}^3## mod the symmetries of an icosahedron described the shape of the universe (near the end of the video).
My question is...
I will be doing a presentation on some knot theory stuff next semester (graduate seminar), and also studying for our Topology qualifier and taking Algebraic topology. My textbook for topology is Munkres (of course!) and the book I am studying knot theory from is Colin Adams wonderful work "The...
I am refreshing my understanding of homology theory (well, recreating from scratch really!) after a thirty year break and there's something that bugs me in how the texts I've seen write about relative homology.
The relative homology module ##H_q(X,A)## is defined as ##ker\ \partial_q^A/Im\...
I have a basic (very basic :)) understanding of the elements of algebra and many years ago I did a course in analysis ... and I would very much like to read my way to an understanding of algebraic topology ..
I figured I should start with some basic texts on topology that (hopefully) head...
I'm a phyiscs student and I have been looking at these lectures:
https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
But I have never learned anything about topology before and was he covers doesn't look like the Topology chapter in my mathematical physics book. I was looking for...
Author: Raoul Bott, Loring Tu
Title: Differential Forms in Algebraic Topology
Amazon Link: https://www.amazon.com/dp/1441928154/?tag=pfamazon01-20
Prerequisities: Differential Geometry, Algebraic Topology
Level: Grad
Table of Contents:
Introduction
De Rham Theory
The de Rham Complex...
I was working on some algebraic topology matters, thinkgs like the connected sum of some surfaces is some other surface. And for this study, I was using the Munkres's famous textbook 'Topology' the algebraic topology part. My qeustions are as follows:
Q1) Munkres introduces 'labelling scheme'...
I am reading Munkres book on Topology, Part II - Algegraic Topology Chapter 9 on the Fundamental Group.
On page 348 Munkres gives the following Lemma concerned with the homomorphism of fundamental groups induced by inclusions":
" Lemma 55.1. If A is a retract of X, then the homomorphism...
On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachement giving Munkres pages 333-334)
"Suppose that h: X \rightarrow Y is a continuous map that carries the point x_0 of X to the point y_0 of Y.
We denote this fact by writing:
h: ( X...
Hi everybody.
Next year I will start an undergraduate course on algebraic topology.
Which book would you suggest as a good introduction to this matter ?
My first options are the following:
1.- "A First Course in Algebraic Topology" by Czes Kosniowski
2.- "Algebraic Topology: An...
Hey guys,
I want to study algebraic topology on my own. I just finished a semester of pointset topology and three weeks of algebraic topology. We did not use a textbook. Can anyone recommend a book on algebraic topology?
Hatcher is fine but it is not as rigorous as I want. Munkres has...
I must say thusfar I read through chapter one of May's book and chapter 0 of Hatcher's, May is much more clear than Hatcher, I don't understand how people can recommend Hatcher's text.
May is precise with his definitions, and Hatcher's writes in illustrative manner which is not mathematical...
Hi, I am enrolled in an Msc programme in pure maths, I wanted to ask for your recommendations on taking a basic graduate course in Algebraic Topology.
Basically my interest spans on stuff that is somehow related to analysis, geometry or analytic number theory.
The pros for choosing this...
Hi all!
I haven't posted here in some time, and I am in need of the expertise of you fine folks. I am busy doing some work on spin geometry. Now, as you guys know, spin structures exist on manifolds if their second Stiefel-Whitney class vanishes. This class is an element of the second...
I 've been reading about Homotopy , homology and abstract lie groups and diff.forms and I would like to see those beautiful ideas applied on a Nonabelian Gauge Theory . Any recommendations for a textbook that apply these ideas to gauge theory ? Text books on particle Physics and QFT do not...
Recently a professor recommended Bott & Tu's Differential Forms in Algebraic Topology to me. My knowledge of algebraic topology is at the level of Munkres' book. Would Bott & Tu's book be too advanced for me to understand at this stage?
Homework Statement
Find an inclusion map i from S^1 to RP^2 such that the induced map of the inclusion (by the fundamental group) is not the zero element.
Known:
pi_1(S^1) = Z and pi_1(RP^2) = Z/2Z
Homework Equations
Can we define i as a composite of two other inclusions?
The...
This is a qualifier exam question in algebraic topology:
Let Z * Z_2 = <a, b | b^2> be represented by X = S^1 \vee RP^2 , i.e. the wedge of S^1 (the unit circle)
and RP^2 (the real projective plane).
For the subgroup H below construct the covering space ˜X by sketching a good picture for...
How do you compute the Fundamental group of the 1-skeleton of the 3-cube I^{3} = [0,1]^{3} ? What about the Fundamental group of the 1- skeleton of the 4-cube I^{4} ?
I know the Fundamental group of a space X at a point x_{0} is the set of homotopy classes of loops of X based at x_{0} . And...
i don't know if i can post it here, like this man https://www.physicsforums.com/showthread.php?t=397395, there's a lot of usefull comment for me.
anyway, I'm still don't really know which one i like, either algebraic topology, or algebraic geometry. but i really do like algebra... so I'm...
Does anyone know of a modern book on algebriac topology developed in a purely categorical framework? I've been recommended Eilenberg and Steenrod (which I may end up getting regardless), but I'm looking for more recent developments in both material and pedagogy.
I see that there are four different GTM textbooks on the subject. Which one of these is the most suitable for self-study?
GTM 56: Algebraic Topology: An Introduction / Massey
GTM 127: A Basic Course in Algebraic Topology / Massey
GTM 153: Algebraic Topology / Fulton
I want to pick up...
Hey guys, i am studying cohomology by hatcher's. Could anyone provide me some ideas on these problems? Thank you all!
Let f : S2n-1 -> Sn denote a continuous map. Let Xf = D2n union f Sn be the space obtained by attaching a 2n-dim cell to Sn using the map f.
i). Calculate the integral...
Hey guys, i am studying cohomology by hatcher's. Could anyone provide me some ideas on these problems? Thank you!
Let f : S2n-1 -> Sn denote a continuous map. Let Xf = D2n union f Sn be the space obtained by attaching a 2n-dim cell to Sn using the map f.
i). Calculate the integral...
In the proof of Proposition 3A.5 in Hatcher p.265 (http://www.math.cornell.edu/~hatcher/AT/ATch3.4.pdf), at the bottow of the page, he writes,
"Since the squares commute, there is induced a map Tor(A,B) -->Tor(B,A), [...]"
How does this follow? The map Tor(A,B)-->A\otimes F_1 is the connecting...
In Hatcher, p. 262 (http://www.math.cornell.edu/~hatcher/AT/ATch3.4.pdf), he writes, just before Lemma 3A.1, "the next lemma shows that this cokernel is just H_n(C)\otimes G. I can't say that I see how this follows.
Thanks!
Hi everyone!
I would like to solve some questions:
Classify up to isomorphism the four-sheeted normal coverings of a wedge of circles. describe them.
i tried to to this and it is my understanding that such four sheeted normal coverings have four vertices and there are loops at each of...
I am trying to show that the space Cone(L(X,x)) is homeomorphic to P(X,x)
where L(X,x) = {loops in X base point x} and
P(X,x) = {paths in X base point x}
I firstly considered (L(X,x) x I) and tried to find a surjective map to P(X,x) that would quotient out right but i couldn't seem to find...
Hey, can anyone help me with this please. I am doing algebraic topology and am particularly stuck on exact sequences. I "understand" the idea of the definition for example:
0\rightarrow A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0
in this short exact...