I want to construct a cell complex that has fundamental group $\mathbb{Z_2}\oplus\mathbb{Z_3}$.
Is there an algorithm to construct a cell complex from given fundamental group?
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3Do you know a space which has fundamental group $\mathbb{Z}_k$? – Michael Albanese Jan 08 '21 at 19:19
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@MichaelAlbanese My guess is it would be k-gon with possible identifications on each edge? – Jan 08 '21 at 19:22
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2Remember $\pi_1(M\times N)\cong \pi_1 M \times \pi_1 N$ – Mr. Brown Jan 08 '21 at 19:23
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@ToInfinity: Do you know any spaces with fundamental group $\mathbb{Z}_2$? – Michael Albanese Jan 08 '21 at 19:30
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@MichaelAlbanese Yes, $\mathbb{R}P^2$. – Jan 08 '21 at 19:31
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2@ToInfinity: Right. Just so I make sure we're on the same page, how do you know that the fundamental group of $\mathbb{RP}^2$ is $\mathbb{Z}_2$? – Michael Albanese Jan 08 '21 at 19:39
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1https://en.m.wikipedia.org/wiki/Presentation_complex – Qiaochu Yuan Jan 08 '21 at 19:47
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@MichaelAlbanese I used Van Kampen to see the fundamental group of $\mathbb{R}p^2$. – Jan 08 '21 at 19:56
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3@ToInfinity: OK. So from van Kampen picture, you can see that $\mathbb{RP}^2$ can be obtained by attaching a disc to a circle by a degree two map on the boundary. Think about what would happen if you change the degree of the attaching map. – Michael Albanese Jan 08 '21 at 19:58
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@MichaelAlbanese I don't quite get it yet. Could you explain little bit more? I appreciate your help. – Jan 08 '21 at 20:13
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@ToInfinity: How are you viewing $\mathbb{RP}^2$? – Michael Albanese Jan 08 '21 at 20:50
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@MichaelAlbanese I use this identification on the first to page (https://en.wikipedia.org/wiki/Real_projective_plane) when thinking about the real projective plane. But I don't think I have a good intuition of that space. – Jan 08 '21 at 21:01
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1@ToInfinity: For the purposes of this question, I think it is better to consider the identification given in this answer. – Michael Albanese Jan 08 '21 at 21:24