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Questions tagged [algebraic-topology]

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
63
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6 answers

What is the difference between homotopy and homeomorphism?

What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic? Otherwise, is there any counterexample?…
liufu
  • 691
49
votes
4 answers

Meaning of relative homology

It is a bit easier to understand the homology $H_1(X, \mathbb Z)$ for various compact surfaces in analogy with handles and so on. There seems to be a nice intuitive picture with handles, holes, etc to think of the first homology group, and similar…
Quinn Rogan
48
votes
5 answers

Fundamental group of the special orthogonal group SO(n)

Question: What is the fundamental group of the special orthogonal group $SO(n)$, $n>2$? Clarification: The answer usually given is: $\mathbb{Z}_2$. But I would like to see a proof of that and an isomorphism $\pi_1(SO(n),E_n) \to \mathbb{Z}_2$ that…
Meneldur
  • 1,509
44
votes
1 answer

Why is a covering space of a torus $T$ homeomorphic either to $\mathbb{R}^2$, $S^1\times\mathbb{R}$ or $T$?

I know a sketch of the proof. M. A. Armstrong 's Basic Topology says that Suppose $X$ has a universal covering space, and denote it by $\tilde{X}$. Then the covering transformations form a group isomorphic to the fundamental group of $X$. Given any…
Roun
  • 3,017
41
votes
2 answers

Homology of cube with a twist

Take the quotient space of the cube $I^3$ obtained by identifying each square face with opposite square via the right handed screw motion consisting of a translation by 1 unit perpendicular to the face, combined with a one-quarter twist of its face…
Juan S
  • 10,268
39
votes
2 answers

The homology groups of $T^2$ by Mayer-Vietoris

If I choose two open sets $A$ and $B$ as depicted on Wikipedia here: then I have an isomorphism between $H_n(A \cap B)$ and $H_n(A) \oplus H_n(B)$ because the two tubes in $A \cap B$ are disjoint. OK, so far so good. Then I write down the…
Rudy the Reindeer
  • 46,359
36
votes
3 answers

Real projective space, the quotient map and covering projection

As I understand it, real projective space $\mathbb R \mathbb P^n$ is defined to be the quotient $S^n / \sim$, where $x \sim y$ iff $x = \pm y$. In other words, elements of $\mathbb R \mathbb P^n$ are pairs of antipodal points on $S^n$. We have the…
Matt
  • 2,343
35
votes
1 answer

Perturbation trick in the proof of Seifert-van-Kampen

The theorem of Seifert-Van-Kampen states that the fundamental group $\pi_1$ commutes with certain colimits. There is a beautiful and conceptual proof in Peter May's "A Concise Course in Algebraic Topology", stating the Theorem first for groupoids…
Martin Brandenburg
  • 163,620
31
votes
4 answers

The circle bundle of $S^2$ and real projective space

Today I felt like computing the integral cohomology of the unit circle bundle of the tangent bundle of $S^2$. For completeness, it is defined by $SS^2=\{x\in TS\colon ||x||=1\}$, where we use the standard Riemannian metric on $S^2$. The cohomology…
Thomas Rot
  • 10,023
30
votes
2 answers

Difference between Deformation Retraction and Retraction

I am currently reading through Hatcher's Algebraic Topology book. I am having some trouble understanding the difference between a deformation retraction and just a retraction. Hatcher defines them as follows: A deformation retraction of a space $X$…
Tyler Clark
  • 1,742
28
votes
3 answers

is the group of rational numbers the fundamental group of some space?

Which path connected space has fundamental group isomorphic to the group of rationals? More generally, is every group the fundamental group of a space?
MBL
  • 972
28
votes
1 answer

Fundamental group of the connected sum of manifolds

Let $M$ and $N$ be two connected manifolds of the same dimension $n$. What is the fundamental group of their connected sum $M \# N$ in terms of $\pi_1(M)$ and $\pi_1(N)$ ?
Adam
  • 289
26
votes
2 answers

Euler characteristic of a boundary?

Why is the Euler characteristic of a boundary even? How can one prove this and is there an geometric way to think about it?
pki
  • 3,743
26
votes
5 answers

Wedge sum of circles and the Hawaiian Earring

The (countably infinite) wedge sum of circles is the quotient of a disjoint countable union of circles $\coprod S_i$, with points $x_i\in S_i$ identified to a single point, while the Hawaiian earring/infinite earring H is the topological space…
Beginner
  • 10,836
26
votes
2 answers

Is homology determined by cohomology?

I am aware of the universal coefficients theorem for cohomology which implies that the homology groups completely determine the cohomology groups. I am wondering if cohomology determines homology in a similar sense? If two spaces have the same…
user101010
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