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

Exhibit a map $X \to K(\pi_1(X),1)$?

Is there a natural map $X \stackrel{f}{\to} K(\pi_1(X),1)$? Can you tell me how it is constructed and what properties has the map $\pi_1(f)$?
Ivan Di Liberti
  • 4,881
3
votes
0 answers

Homology group of the complement of an euclidean space and its subspace

I'm considering the following question - if M is an m-dimensional subspace of $\mathbb{R}^n$, then how to compute the homology of $\mathbb{R}^n - M$. Thanks!
Riley Zhang
  • 101
3
votes
1 answer

fundamental group of $\Bbb R^3 \setminus$ (two linked circles)

Let $ X = \mathbb R^3 \setminus A$, where $A$ represents two linked circles. I'd like to calculate $\pi_1(X)$, using van Kampen. I don't know how to approach this at all - I can't see an open/NDR pair $C,D$ such that $X = C \cup D$ and $C \cap D$ is…
Abhishek Shrivastava
  • 854
3
votes
2 answers

The fundamental group of the circle with some points identified

I'm thinking about the fundamental group of a circle with some points identified. I mean let $r:\mathbb S^1\to \mathbb S^1$ be a quotient map mapping the point of the circle $(cos \theta, sin \theta )$ to $(cos(\theta+2\pi /n),sin (\theta+2\pi…
user42912
  • 23,582
3
votes
1 answer

Example of $Y = A \cup B \cup C$ such that reduced homology groups are trivial for all intersections but not for $Y$

I'm trying to solve the following problem: Give an example of $Y = A \cup B\cup C$ such that $A,B,C$ are open subsets of $Y$ and the reduced homology groups of $A,B,C, A\cap B, A\cap C, B \cap C, A \cap B \cap C$ are all trivial, the sets $A \cap B,…
NewUserForNow
  • 33
3
votes
3 answers

Exact sequence for homology of space, subspace and quotient space

Hatcher states the following theorem on page 114 of his Algebraic Topology: If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence …
Holdsworth88
  • 8,818
3
votes
0 answers

Klein Bottle with a glued disk on an essential orientation preserving curve. Fundamental groups and coverings

As the title suggests, I'm studying the following topological space: which is a Klein Bottle $K$ with a disk glued along an essential orientation preserving curve. I need to compute its fundamental group and clearly describe all its connected…
Luigi M
  • 3,887
3
votes
1 answer

The wedge sum is a retract of the torus?

I'm trying to prove or disprove that the wedge sum of two circles is a retract of the torus. Intuitively it seems true, because the torus is defined as $S^1\times S^1$. I tried to disprove also, but the only tools I know to do this is if one space…
user42912
  • 23,582
3
votes
0 answers

Coning a manifold into the same dimension

Let $X \hookrightarrow \mathbb R^d$ where $d$ is minimal (that is, $X \not \hookrightarrow \mathbb R^{d-1}$). When $X$ is the $(d-1)$-sphere, then the cone of $X$ still embeds into $\mathbb R^d$. Are there any other manifolds $X$ which have this…
user242594
3
votes
1 answer

Definition of singular homology

Let $R = {\bf Z}$ Let $\partial_i : C_i \rightarrow C_{i-1}$ be a boundary map where $C_{-1} = \{ 0 \}$, $C_i$ is the set of all maps $f$ and $f: \Delta_i \rightarrow M$. Let $Z_n = $Ker of $\partial_n$ If $f $ is a loop in $M$ then…
HK Lee
  • 19,964
3
votes
1 answer

A problem from Hatcher's algebraic topology

This is Ex 1.1.19 from Hatcher's algebraic topology. Show that if $X$ is a path-connected 1-dimensional CW complex with basepoint $x_0$ a 0-cell, then every loop in $X$ is homotopic to a loop consisting of a finite sequence of edges traversed…
JSCB
  • 13,456
  • 15
  • 59
  • 123
3
votes
0 answers

Maps on n-dimensional torus with points removed

I am interested in the homotopy types of maps $m:T^n\setminus \left\lbrace x_0,x_1, ... x_k \right\rbrace \rightarrow S^2$. My first try was to figure out how the n-dimensional torus looks like, if n-points are removed. I know that for the two…
wog87
  • 31
3
votes
1 answer

Cell Structure on Sphere with Two points identified

My question is related to this one here but is different in that I am wondering about the CW structure on such a space. I am trying to put a CW structure on $S^2/S^0$ and I think that we have $1$ 0 -cell, $1$ 1 - cell and $1$ 2-cell. My $1$ - cell…
user38268
3
votes
1 answer

What happens to homology without cycles?

Let X be a topological space and $C_n(X)$ be the singular chain complex. The homology is defined to be $H_n(X)$ = $ ker \partial_n / im \partial_{n+1}$. What happens if we take $ K_n(X) = C_n(X) / im \partial_{n+1}$ instead? (The idea comes from…
user40167
3
votes
1 answer

Hatcher's problem 2.2.39

Use the preceding exercise (2.2.38) to derive relative Mayer–Vietoris sequences for CW pairs $(X,Y) = (A ∪ B,C ∪ D)$ with $A = B$ or $C = D$. I tried to solve this just in the case $A=B$, with two triplet of the type $(X,A,B)$, where $B \subset A…
bpittcher
  • 47
1 2 3
…
99 100