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Problem of the Week #210 - Jun 07, 2016

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Euge

MHB Global Moderator
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Jun 20, 2014
1,892
Here is this week's POTW:

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Evaluate the abelianization of the fundamental group of the $n$-fold connected sum $\underbrace{\Bbb RP^2\, \# \cdots \#\, \Bbb RP^2}_{n}$.
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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Euge

MHB Global Moderator
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Jun 20, 2014
1,892
No one answered this week's problem. This time I'll give someone else's solution -- John Lee's in his book Introduction to Topological Manifolds, on page 267.



Write the fundamental group as $H \cong \langle \beta_1,\ldots, \beta_n\, |\, \beta_1^2\cdots \beta_n^2\rangle$. Let $f$ denote the nontrivial element of $\Bbb Z/2$, and define $\varphi : \operatorname{Ab}(H) \to \Bbb Z^{n-1} \times \Bbb Z/2$ by

$$\varphi(\beta_i) = \begin{cases} e_i, & 1 \le i \le n-1;\\f - e_{n-1} - \cdots - e_1, & i = n\end{cases}$$ By direct computation $\varphi(\beta_1^2\cdots \beta_n^2) = (0,\ldots, 0)$ (noting that $f + f = 0$), so $\varphi$ gives a well-defined map from $H$ that descends to $\operatorname{Ab}(H)$. The homomorphism $\psi : \Bbb Z^{n-1} \times \Bbb Z/2 \to \operatorname{Ab}(H)$ defined by $\psi(e_i) = [\beta_i], \psi(f) = [\beta_1\cdots \beta_n]$ is the inverse for $\varphi$.
 
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