Homotopical syzygies (Q2707510)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Homotopical syzygies |
scientific article |
Statements
23 October 2001
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braid groups
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classifying spaces of discrete groups
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permutohedra
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presentations
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Stasheff polytopes
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cellular complexes
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free resolutions
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Igusa pictures
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identities among relations
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homological syzygies
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homotopy groups
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Homotopical syzygies (English)
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For a group \(G\), presented by generators and relations, two tasks appear:NEWLINENEWLINE-- build a cellular complex \(BG\) with the homotopy type of \(K(G,1)\);NEWLINENEWLINE-- build an explicit free resolution of the trivial \(G\)-module \(\mathbb{Z}\).NEWLINENEWLINENEWLINETo this aim in low dimension, the author introduces homotopy 2-syzygies strongly related with \textit{K. Igusa}'s pictures [K-Theory 7, No. 3, 201-224 (1993; Zbl 0793.19001)], identities among relations [\textit{R. Brown, J. Huebschmann}, Lond. Math. Soc. Lect. Note Ser. 48, 153-202 (1982; Zbl 0485.57001)], and homological syzygies. Then higher homotopy syzygies are defined and compared with higher homological syzygies as well.NEWLINENEWLINENEWLINESeveral examples of groups are treated which fit into this framework. At the end, higher homotopical syzygies for the parametrized braid group are constructed.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00041].
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