Tagged Questions
1
vote
2answers
42 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 ...
7
votes
1answer
99 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
203 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 ...
5
votes
1answer
51 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 ...
9
votes
3answers
196 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
61 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 ...
14
votes
1answer
536 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 ...
4
votes
0answers
116 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 ...
2
votes
1answer
208 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
328 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
812 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 ...
