Mathematics Stack Exchange
Mathematics Stack Exchange

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
3
votes
1 answer

Weakly contractible space(or n-connectedness) and homotopic maps

Lemma: Suppose we have a $n$-dimensional CW complex $X$ and an aspherical space $Y$, where aspherical just means $\pi_n(Y)=0$ for all $n\geq 0$. Let $f,g:X\rightarrow Y$ be continuous maps. Then $f$ and $g$ are homotopic. So my question is how to…
Enigma
  • 3,909
3
votes
2 answers

Given any Group G, is there a topological space whose fundamental group is exactly G?

Given any Group $G$, is there a topological space whose fundamental group is exactly $G$? If yes, what (which Theorem) is this result based on?
alexlo
  • 523
  • 3
  • 13
3
votes
0 answers

Seifert van Kampen proof for open coverings.

In my lecture we proofed the "easy" seifert van kampen theorem namly: If $X=U\cup V$ U,V open and U,V,$U\cap V$ path connected. Let $x_0\in U\cap V$ then $\pi_1(X,x_0)\cong\pi_1(U,x_0)*_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0)$. where the amalgamation is…
Philip Mayer
  • 351
3
votes
1 answer

Basic definition of Cellular Boundary Formula

I'm working thru Hatcher on my own and am having trouble with basic definitions relating to his Cellular Boundary Formula on p 140. What is the domain and range of the function in question? Do the d(.)expressions refer to the composite boundary…
PossumP
  • 1,734
  • 13
  • 24
3
votes
0 answers

Allen Hatcher Algebraic Topology Section 1.3 Exercise 27 (1.3.27)

For a universal cover $p:\widetilde{X}\to X$ we have two actions of $\pi_1(X,x_0)$ on the fiber $p^{-1}(x_0)$, namely the action given by lifting loops at $x_0$ and the action given by restricting deck transformations to the fiber. Are these…
khalatnikov
  • 815
3
votes
1 answer

What does it mean to say that a pair of points are antipodal in a topological sphere?

A pair of points are antipodal if they are diametrically opposite to each other. This definition makes perfect sense when one thinks of the unit 2-sphere centered at the origin and embedded in $R^3$; that is, the set of all points for which…
anonmath
  • 33
3
votes
1 answer

Prove that $\mathbb{R}P^2\ \# \ K$ and $\mathbb{R}P^2\ \# \ \mathbb{T}^2$ are homeomorphic.

On a introductory Topology course, I learned how to make some connected sums of spaces informally. For example, in the figure below, we have a draft for $\mathbb{R}P^2\ \# \ \mathbb{R}P^2$, where $\mathbb{R}P^2$ is the real projective plane. First,…
rgm
  • 817
3
votes
1 answer

What does a linear map induce on cohomology rings of associated projective spaces?

Suppose $f:\mathbb{R}^n\setminus\{0\}\to \mathbb{R}^m\setminus\{0\}$ is a linear map inducing a map $g:\mathbb{R}P^{n-1}\to \mathbb{R}P^{m-1}$ and hence a homomorphism $$ g^*:\mathbb{Z}/2\mathbb{Z}[x]/(x^m)\to \mathbb{Z}/2\mathbb{Z}[y]/(y^n) $$ on…
user398460
  • 55
3
votes
3 answers

Fundamental group of a triangle with the three vertices identified?

Which is the fundamental group of a triangle with the three vertices identified? I call $X$ the previous space. I take $U$ be a disk inside the triangle and $V$ the complement of a disk inside the triangle (I take them so that $U \cap V$ is homotopy…
TheWanderer
  • 5,126
3
votes
1 answer

Using Van-Kampen theorem to find a fundamental group

I would be grateful if someone takes a look at my solution to the following problem to see if it is correct. Thanks a lot for helping people even during the holidays. Sorry, I have to delete the question due to some moral stuff.
Aleph-null
  • 363
  • 2
  • 14
3
votes
1 answer

Fundamental group of knot complement

I would be happy if one can help me with how to approach this problem. Suppose that $K$ is a knot in $\mathbb{R}^3$, show that $\pi_1(\mathbb{R}^3 \setminus K)$ is isomorphic to $\pi_1(\mathbb{S}^3 \setminus K)$. I know I need to use Van-Kampen's…
Aleph-null
  • 363
  • 2
  • 14
3
votes
1 answer

Collapsing the boundary of a Möbius Strip to a point

I'm strunggling to prove that when we collapse the boundary of a Möbius strip we obtain the RP² thanks
user42912
  • 23,582
3
votes
1 answer

How many times does $8z^4+4z^3+2z^2+z^{-1}$ wind around $0$?

I am a newbie to abstract topology and am working through some exercises regarding fundamental groups. Here is the problem. Consider $f: S^1 \to \mathbb{C} \setminus \{ 0 \}$ given by $z \mapsto 8z^4+4z^3+2z^2+z^{-1}$. What is the winding number of…
Minsheng Liu
  • 331
3
votes
2 answers

Finite graph homotopically equivalent to wedge sum of finite circle.

Show that if X is a finite graph i.e. a graph with finitely many vertices and finitely many edges, then X is homotopically equivalent to wedge sum of finitely circles. I know that I will be able to make the connected components of finite graph…
scot
  • 31
3
votes
1 answer

Deformation retraction induced on quotient space

I'm trying to show that a punctured torus deformation retracts to a wedge of 2 circles. So I considered a punctured solid square (which eventually becomes a torus after identification of opposites sides), and make it deformation retract onto its…
Tom
  • 1,125
1 2 3
…
99 100