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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
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4 answers

Hausdorff condition for CW complexes

Consider the definition of CW complex from wikipedia. It is assumed that the space is Hausdorff. Are there problems if we drop this assumption? What is an example of a space satisfying all the CW complex axioms except this condition?
Cerolein
  • 91
4
votes
0 answers

Adjointness of homological and cohomological transfer maps

Suppose $p: \tilde X \to X$ is a normal covering space with deck group $\Gamma$. Then there are transfer maps $$ \tau^*: H^*(\tilde X, \mathbb Q) \to H^*(X, \mathbb Q) $$ and $$ \tau_*: H_*(X,\mathbb Q) \to H^*(\tilde X, \mathbb Q). $$ (Suppose I…
NKS
  • 4,392
4
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1 answer

Singular complex is a delta complex

If I understand correctly, $\Delta$-complex on a space $X$ is defined to be a collection $\Delta(X)$ of cont. funtions $\sigma:\Delta^n\to X$ such that: 1) restriction of $\sigma$ to any face is in $\Delta(X)$ 2) restriction of $\sigma$ to the…
user129686
4
votes
1 answer

Determining the generators of cohomology (as a ring)

I am working on a problem to show that the cohomology graded rings of $\mathbb{C}P^3$ and $S^2$ x $S^4$ are not isomorphic (unreduced with integer coefficients) I have already calculated the graded groups of both and shown that they are isomorphic…
Elliot
  • 817
4
votes
3 answers

Why is this a wrong Triangulation?

Why is this a wrong Triangulation ? I have to say, we had triangulation at the end of the topology course, so not in details. And the professor only mentioned the basic rules for the triangulation, but in this case, idon't know why it fails.
derivative
  • 2,450
4
votes
1 answer

Homotopy equivalence from torus minus a point to a figure-eight.

Let $I$ be the unit interval, and define the torus using the usual identifications on $I \times I$. I've shown that $I \times I - \{x\}$ (where $x$ is a point not on the boundary of $I \times I$) is homotopy equivalent to its boundary (and that it…
user121410
  • 329
4
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1 answer

Fundamantal group of a regular covering space

Let $B$ be the space of figure $\infty$ (with $x$(the red circle) and $y$(the black one) as generators) and $E$ its covering space (in the picture below). let $P_{*}: \Pi(E,a) \to \Pi(B,b) $ be the covering map which transformes $a_1 \to x$,$a_2 \to…
Eli Elizirov
  • 1,820
4
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1 answer

$X$ a CW complex is contractible if it's the union of an increasing sequence

I'm doing exercise 11 on page 358 in Hatcher: Show that a CW complex is contractible if it is the union of an increasing sequence of subcomplexes $X_1 \subset X_2 \subset \dots $ such that each inclusion $X_i \hookrightarrow X_{i + 1}$ is…
Rudy the Reindeer
  • 46,359
4
votes
1 answer

$H_1(X,A) = 0 \iff H_1(A) \rightarrow H_1(X)$ surjective and $X_i$ contains no more than one path-component of $A$

I've done another exercise in Hatcher and was wondering if you could tell me if I did it right. The exercise: $H_1(X,A) = 0 \iff H_1(A) \rightarrow H_1(X)$ surjective and $X_i$ contains no more than one path-component $A_i$ of $A$ where $X_i$ are…
Rudy the Reindeer
  • 46,359
4
votes
2 answers

$H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$

I've been doing some more exercises in Hatcher, in particular the following: Show that $H_0(X,A) = 0$ iff $A$ meets each path-component of $X$. "$\Leftarrow$": Let $x_i \in A \cap X_i \neq \emptyset \forall $ path-components $X_i$. Then for any $x…
Rudy the Reindeer
  • 46,359
4
votes
2 answers

What is the topology of a simplicial complex?

I know what a simplicial complex is, but when reading about triangulations on surfaces I found that there must exist a homeomorphism betwen the space underlying the surface and some simplicial complex. So my question is, how is defined the topology…
Mary Wills
  • 41
4
votes
1 answer

Hawaiian earring is not a cofibration

Let $X$ be the union of all circles centered at $(0,\frac{1}{n})$ with radius $\frac{1}{n}$ for $n\in N$. Let $A$ be $(0,0)$. Show that $A\to X$ is not a cofibration. This appears as a non-example for inclusion != cofibration. However I am not able…
Ma Ming
  • 7,482
4
votes
1 answer

induced maps in homology

Let $S^1=I^1/ \partial I^1$, where $I^1=[0,1]$ with base point ${0}$ and $I^n=I^1\times I^1 \times \dots \times I^1$ (n times). $S^p=S^1 \wedge \dots \wedge S^1 $ where $\wedge$ is the smash product. Now we define $\gamma_1:I^1 \to S^1$ to be the…
Rungo
  • 490
4
votes
1 answer

Any two paths in $X = \mathbb{R}^n$ having same initial and end point are homotopic

Suppose $X = \mathbb{R}^n$. Let $\gamma, \alpha : [0,1] \to X $ be to paths such that $\gamma(0) = \alpha(0) = x_0 , \; \; \gamma(1) = \alpha(1) = x_1$. We want to show $\gamma$ and $\alpha$ are homotopic. My try: Take $F(s,t) = f_t(s) =…
ILoveMath
  • 10,694
4
votes
1 answer

Mapping cone not homotopy equivalent to quotient space

My question is about a "non-example" to theorem 1.6 in chapter VII in Bredon. We have an inclusion $i: A \to X$, with $A = \{0\} \cup \{1/n | n = 1,2,...\}$, and $X = [0,1]$. Then $X/A$ is a one-point union of an infinite sequence of circles with…
Balerion_the_black
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