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How to construct a surface of genus $g$ by identifying sides of a $4g$-gon?

Since you asked for an explanation for 2 and 3 (and not 1) I'll assume you understand the basic idea of identifications, and will try to walk you through the process of identifying. Excuse the poor ...
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• 11.7k
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An infinite dimensional CW complex always has infinitely many non-trivial homology groups?

A counterexample is $S^\infty$, the union of $S^n$ for all $n\geq 0$. This is a complex with two cells in each dimension, so it is infinite dimensional, but it is contractible.
• 58.3k
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• 332k

Compact subset of CW complex

There are some important theorems about compact sets that can simplify your work: If $A$ is closed, $A\subseteq B$, and $B$ is compact, then $A$ is compact If $A$ is compact and $f:A\to B$ is ...
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CW structure of the universal cover

When you have a covering map $p:\tilde{X}\to X$ with a CW structure on $X$, you can use the homotopy lifting property to lift each cell of $X$ to a collection of cells of $\tilde{X}$, since you can ...
• 19.5k
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• 5,811
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Covering $\Bbb RP^\text{odd}\longrightarrow X$, what can be said about $X$?

If $n = 1$, then $\mathbb{RP}^1 = S^1$ which only covers itself. If $n > 1$, the manifold $\mathbb{RP}^{2n-1}$ covers infinitely many manifolds which are pairwise non-homotopy equivalent. To see ...
• 100k

Showing Hawaiian earrings are not CW complexes

Here's another way to show it. As you say, if $X$ were a CW-complex, then it would have to be a finite CW-complex since it is compact. Since every finite CW-complex has finitely generated homology, ...
• 332k
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Gluing of Möbius strips

I am not sure why the comment is upvoted: your meaning is clear, take $M_1 = \cdots = M_n = M$ all copies of the Mobius band; there are inclusion maps $f_i: S^1 \to M_i$ for all $i$ parameterizing the ...
• 86
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Examples of isomorphic group presentations with homotopy NONequivalent complexes

Take $$G=\langle x,y\mid x^2=y^3\rangle.$$ This is the standard presentation for the fundamental group of the trefoil knot. Of note is the fact that this group is Hopfian and non polycyclic-by-finite. ...
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On a cw complex structure on the $2$- sphere
$\bullet$ For $v=1$ we have $f=1+e$. So, consider a wedge of $e$-many circles on $\Bbb S^2$ with wedge point as $v$, and then fill up the complement of this wedge by $(1+e)$ many $2$-cells. $\bullet$ ...