Comparison of the geometric bar and \(W\)-constructions. (English) Zbl 0924.18007
For a simplicial group \(K\), one can form the canonical simplicial \(K\)-bundle \(WK \to \bar{W}K\) and then the geometric realization \(| WK| \to | \bar{W}K| \). Alternatively, one can form the realization \(| K| \) of \(K\) and the \(| K| \)-bundle \(E| K| \to B| K| \) of Milgram and Steenrod. The initial observation is that these constructions are isomorphic if \(K\) is a discrete group: indeed, this was surely the motivation for the definition of the functors \(E\) and \(B\). The main result of this paper is that, in fact, the two constructions are naturally isomorphic for all \(K\). The proof is rather elegant: the authors argue that both \(| WK| \) and \(E| K| \) are built as \(| K|\)-spaces in a recursive fashion via simple functors and that, furthermore, the initial data are the same in both cases; hence, the two must be isomorphic. They point to a paper by the second author for applications: see J. Huebschmann [“Extended moduli spaces, the Kan construction, and lattice gauge theory”, Topology 38, No. 3, 555-596 (1999)].
Reviewer: Paul G.Goerss
MSC:
| 18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |
| 55P35 | Loop spaces |
| 55U10 | Simplicial sets and complexes in algebraic topology |
| 55Q05 | Homotopy groups, general; sets of homotopy classes |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
Citations:
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