2
votes
0answers
30 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
4
votes
0answers
56 views

Contractible Subspace and Homotopy Equivalence

It is known that if pair $(X,A)$ has the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence. I am wondering what if the homotopy ...
1
vote
1answer
28 views

$H^1(X) = [X,\mathbb{T}]$?

This is a stupid question, but here goes. I have a compact Hausdorff space $X$, and I am talking about $[X,\mathbb{T}]$, the group of homotopy classes of maps $X \to \mathbb{T}$, where $\mathbb{T}$ ...
3
votes
0answers
94 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathcal{Top}$, $k$-closed subset $Y\subset X$, means $u^{-1}(Y)$ is closed in $C$ for any $u: ...
3
votes
2answers
77 views

For any element $e$ of an open set $V$ of a covering space, does there exist a sheet $S$ such that $e\in S\subseteq V$

Let $p:E\rightarrow X$ be a covering map. Let $V$ be any open subset of $E$ and $e$ be any element of $V$. I feel that the following statement must be true: There exists an evenly covered open subset ...
11
votes
2answers
323 views

A comparison between the fundamental groupoid and the fundamental group

Are there two path connected topological spaces $X,Y$ such that the fundamental groupoid of $X$ is not isomorphic to the fundamental groupoid of $Y$ but the fundamental group of $X$ is isomorphic to ...
1
vote
2answers
112 views

example of homotopy which is not path homotopy

Can someone give me a simple, concrete example of a homotopy, which is not a path homotopy? Let $f, f'$ be continuous maps from $X$ to $Y$, and let $F: X \times I\to Y$ a continuous map such ...
8
votes
1answer
238 views

what's the role of fiber bundles play in understanding the base space?

Suppose $\pi: E\to B$ be a fiber bundle, and assume $B$ is connected, I want to know: 1, What do the fiber bundles (e.g., the covering spaces) on $B$ help understand the topology, or the structure on ...
8
votes
2answers
316 views

Is every open, bounded and simply connected subset of $\mathbb{R}^n$ essentially a ball?

Let $\Omega\subset \mathbb{R}^n$ be open, bounded and simply connected. I wonder if the answer to the following question is known: Is there a homeomorphism $\Omega\to \operatorname{B}_1(0)$, where ...
6
votes
1answer
122 views

Example of a pair of non-cobordant manifolds

So far, any source I consult will gladly talk about cobordism classes of closed (compact and without boundary) oriented manifolds, but I have yet to see an example of a pair of manifolds which are not ...
11
votes
3answers
308 views

Is every contractible space a cone?

It is easy to show that for any topological space $X$, the cone $CX$ is contractible. I am interested in the converse. If $Y$ is a contractible space, is $Y$ homeomorphic to $CX$ for some topological ...
1
vote
1answer
215 views

Example of a non-injective retract induced homomorphism of fundamental groups

When answering this question I used the fact that when we have a retract $r:X \rightarrow Y$ the induced homomorphism $r_\ast: \pi_1(X) \rightarrow \pi_1(Y)$ is surjective. I can recall how to prove ...
3
votes
0answers
123 views

Relative Homology and Quotients

Are there any topological spaces $X$ with subspaces $A$ such that $H_n(X,A)$ is not isomorphic to $H_n(X/A)$? I've been trying some familiar spaces, but everything seems to be me an isomorphism ...
14
votes
1answer
759 views

A counterexample in topology

Semi-local simple connectedness is a property that arises in Algebraic Topology in the study of covering spaces, namely, it is a necessary condition for the existence of the universal cover of a ...
5
votes
0answers
184 views

A counter example in obstruction theory

Let $K$ denote a simplicial complex and $Y$ some topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation: There is a map ...
4
votes
1answer
435 views

Spaces with equal homotopy groups but different homology groups?

Since it's fairly easy to come up with a two spaces that have different homotopy groups but the same homology groups ($S^2\times S^4$ and $\mathbb{C}\textrm{P}^3$). Are there any nice examples of ...
6
votes
4answers
702 views

Two CW complexes with isomorphic homotopy groups and homology, yet not homotopy equivalent

A standard example of two CW complexes which have isomorphic homotopy groups but are not homotopy equivalent is $ RP^2 \times S^3$ and $RP^3 \times S^2$. The easiest way to see that they are not ...
15
votes
2answers
1k views

How useless can the Mayer-Vietoris sequence be in general?

In an algebraic topology course I'm taking we are often asked to compute the homology groups of a space $X = A \cup B$ using the Mayer-Vietoris sequence, and it happens in all of the examples I've ...