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.

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  1. Infrared

    Challenge Math Challenge Thread (October 2023)

    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 Do not cite theorems that trivialize the problem you're solving. 3.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...
  2. fibrebundle

    Looking to meet other people who like math

    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...
  3. A

    What benefits can this site offer to increase knowledge and understanding?

    Hi all, It is nice to be a member in this site! Hope it will be beneficial and add to my knowledge and understanding.
  4. K

    I Showing that a group acts freely and discretely on real plane

    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...
  5. S

    A Chain Rule for Pushforwards

    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∗...
  6. T

    B Algebraic Topology in the tv show The Big Bang Theory

    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...
  7. N

    I Equivalence of Covering Maps and Quotient Maps

    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...
  8. Auto-Didact

    A Algebraic topology applied to Neuroscience

    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...
  9. KennethK

    How to show simplicial complex is Hausdorff?

    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?
  10. L

    A Very basic question about cohomology.

    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...
  11. G

    A Period matrix of the Jacobian variety of a curve

    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...
  12. L

    A Why the Chern numbers (integral of Chern class) are integers?

    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...
  13. V

    I Can a CW complex exist without being a Hausdorff space?

    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...
  14. Jianphys17

    Best book for undergraduate study algebraic topology

    In your opinion what is the best book for a first approach to algebraic topology, for self studt more properly!
  15. P

    I Triangulation of circle and disk in R2

    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...
  16. I

    Courses Representation theory or algebraic topology

    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...
  17. 1

    A The fundamental group of preimage of covering map

    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...
  18. N

    What shape does SO(3)/A5 describe and how can it be visualized?

    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...
  19. dkotschessaa

    Knot Theory & Topology w/ Munkres & "The Knot Book" by Adams

    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...
  20. andrewkirk

    Abuse of notation in relative homology theory

    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\...
  21. Math Amateur

    MHB What are some recommended texts for studying algebraic topology?

    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...
  22. W

    Should I learn Algebraic Topology?

    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...
  23. N

    [algebraic topology] proving local degree is +1

    Homework Statement Suppose we have f: \mathbb C^n \to \mathbb C^n with f(z_1, \cdots,z_n) = (\sigma_1(\mathbf z), \cdots, \sigma_n(\mathbf z)) where the sigmas are the elementary symmetric functions (i.e. \sigma_1 = \sum z_j \quad \sigma_2 = \sum_{i < j} z_i z_j \quad etc) Note if we look...
  24. micromass

    Topology Differential Forms in Algebraic Topology by Bott and Tu

    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...
  25. J

    Algebraic Topology: Connected Sum & Reference Help

    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'...
  26. Math Amateur

    Algebraic Topology - Retractions and Homomorpisms Induced by Inclusions

    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...
  27. Math Amateur

    Algebraic Topology - Fundamental Group and the Homomorphism induced by h

    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...
  28. Karlx

    A good book for an introduction to Algebraic Topology

    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...
  29. R

    What Are Your Recommendations for Algebraic Topology Textbooks?

    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...
  30. MathematicalPhysicist

    Hatcher Vs. May's Algebraic Topology.

    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...
  31. MathematicalPhysicist

    Courses Taking a graduate course in Algebraic Topology or not?

    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...
  32. S

    ALgebraic Topology Query (Hatcher) - Not Homework

    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...
  33. Z

    Applying Algebraic Topology, Geometry to Nonabelian Gauge Theory

    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...
  34. P

    How accessible is Bott & Tu's book on algebraic topology?

    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?
  35. K

    Algebraic Topology - Inclusion into RP^2

    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...
  36. Y

    Constructing Covering Spaces for Algebraic Topology Qualifier Exam Question

    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...
  37. M

    Algebraic Topology: Fundamental group of a cube

    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...
  38. A

    Differences between Algebraic Topology and Algebraic Geometry

    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...
  39. R

    Algebraic Topology via Categories

    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.
  40. P

    What's the difference between differential topology and algebraic topology?

    Having some knowledge of differential geometry, I want to self-study topology. Which of the two areas shall I study first? Thanks for answer!
  41. H

    Which algebraic topology textbook is the best for self-study?

    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...
  42. 2

    Problem in Algebraic Topology

    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...
  43. 2

    How Do You Calculate Homology in Algebraic Topology Problems?

    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...
  44. A

    Edwin Spanier Algebraic Topology difficulty?

    How difficult is Spanier's Algebraic Topology text to understand? How about the exercises?
  45. quasar987

    Another algebra question in algebraic topology

    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...
  46. quasar987

    Algebra question in algebraic topology

    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!
  47. K

    Algebraic topology, groups and covering short, exact sequences

    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...
  48. N

    Algebraic Topology: Showing Cone(L(X,x)) is Homeomorphic to P(X,x)

    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...
  49. T

    Constructing Mono/Epi Functions for Algebraic Topology

    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...