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Results tagged with gn.general-topology
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user 39747
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
15
votes
Accepted
Hairy ball theorem for odd-dimensional spheres
The Lefschetz fixed point theorem implies that any $f: S^n \to S^n$ without fixed points has degree $(-1)^{n+1}$. But an even map $S^n \to S^n$ has even degree, since it factors as
$$
S^n \xrightarrow …
- 10.8k
3
votes
Cohomology version of Moore space
As Allen Hatcher answered, there is no space whose cohomology is a countable direct sum of $\mathbb{Z}$'s in a single degree, and the cohomology of a wedge of spheres is instead a product.
However, th …
- 10.8k
1
vote
Can the loops in the definition of the fundamental group be considered injective?
For the new version of the question (where you allow to replace the space by a homotopy equivalent one) the answer is now "yes": just replace every $X$ by $\lvert\operatorname{Sing}(X)\rvert$. This is …
- 10.8k
5
votes
A natural embedding of the total space of tautological bundle over $G(2,n)$ in $G(2,n+1)$
There's two kinds of $k$-planes in $\mathbb{R}^n\times \mathbb{R}$: Those that project isomorphically to $\mathbb{R}^n$, and those that contain $\mathbb{R}$.
The former are given as graphs of linear f …
- 10.8k