5
$\begingroup$

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let $q: D^k \to D^r$ be a map and $r \leq k$. Let $$W = S^k \sqcup D^r/\sim$$ where $S^k \supset D^k \ni x \sim q(x)$ is the equivalence relation (one can also express $W$ it as pushout of the inclusion map and $q$).

$Q:$ Under what conditions on $q$, the space $W$ is homeomorphic to $S^k$?

Edit: All the disks mentioned above are closed disk.

One of the conditions that I think of is if $q^{-1}(pt)$ is contractible for every $pt \in D^r$. Is this a sufficient condition?

Improve this question
$\endgroup$
8
  • $\begingroup$ Your question is equivalent to the following one: for which q, W is a k-manifold, and then you conclude by left properness and rigidity of the k-spheres. $\endgroup$
    – Ilias A.
    Commented Jul 28, 2014 at 22:09
  • $\begingroup$ @Fedotov If you don't mind, can you please expand on the idea? I did not follow the conclusion part. $\endgroup$
    – Prasit
    Commented Jul 28, 2014 at 22:47
  • 3
    $\begingroup$ Since the category of topological space is left proper, the map $f:S^{k}\rightarrow W$ is a weak equivalence. If $W$ is a manifold then by generalized poincare (conjecture, now a theorem) $W$ is homeomorphic to $S^{k}$. So you need to find under which condition on $q$, $W$ is a manifold (actually compact). $\endgroup$
    – Ilias A.
    Commented Jul 28, 2014 at 22:57
  • $\begingroup$ So where is it proved that pushouts of weak equivalences are weak equivalences? $\endgroup$
    – ThiKu
    Commented Jul 29, 2014 at 16:11
  • $\begingroup$ Push-out of a weak equivalence along a cofibration is a weak equivalence in a left proper model category (you can take it as a definition of left properness). $\endgroup$
    – Ilias A.
    Commented Jul 29, 2014 at 17:41

0

Reset to default

You must log in to answer this question.