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Results tagged with 20-questions
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user 75
In the earliest days of MathOverflow, there was a '20 questions' seminar (see <http://sbseminar.wordpress.com/category/20-questions/>) run by graduate students at Berkeley. Many questions from the seminar were cross-posted to MathOverflow. This tag now exists solely for the historical record.
7
votes
What is an example of a topological space that is not homotopy equivalent to a CW-complex?
The Cantor set, or any space X with no nontrivial path which is not discrete. If X did have the homotopy type of a CW-complex, the identity map from X with the discrete topology to X would be a homot …
- 31.2k
28
votes
Accepted
Which sequences can be extended to analytic functions? (e. g., Ackermann's function)
It's a standard theorem in complex analysis that if $z_n$ is a sequence that goes to infinity, there is an entire function taking any prescribed values at the $z_n$. There is a function $f$ vanishing …
- 31.2k
6
votes
Cohomology and Eilenberg-MacLane spaces
Here's a precise statement: reduced singular cohomology $H^n(X;G)$ is naturally isomorphic to homotopy classes of pointed maps from $X$ to $K(G,n)$, for any pointed topological space $X$ having the ho …
- 31.2k
13
votes
Is there a category in which finite limits and directed colimits *don't* commute
Consider the poset of closed subsets of $[0,1]$. Let $a=\{0,1\}$ and $b(r)=[0,r]$ for $r<1$. Then the (directed) colimit of the $b(r)$ is $b=[0,1]$, and the product (i.e., intersection) of $b$ and a …
- 31.2k
8
votes
When does the sheaf direct image functor f_* have a right adjoint?
If $f_\ast$ has a right adjoint, it must preserve colimits and hence be right-exact. Thus a necessary condition is that the higher derived functors vanish. In particular, when everything is affine an …
- 31.2k
60
votes
Accepted
understanding Steenrod squares
Here's one way to understand them. The external cup square $a \otimes a \in H^{2n}(X \times X)$ of $a \in H^n(X)$ induces a map $f:X \times X \to K(Z_2, 2n)$. It can be show that this map factors th …
- 31.2k