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A category of pseudo-tangles with classifying space \(\Omega^\infty S^\infty\) and applications. (English) Zbl 1024.18007

Barratt and Priddy’s theorem states that there is a canonical map \(\coprod_{n\geq 0}B \Sigma_n\to \Omega^\infty S^\infty\) which induces a group completion [cf. S. B. Priddy, Proc. Sympos. Pure Math. 22, 217-220 (1971; Zbl 0242.55011); G. Segal, Topology 13, 293-312 (1974; Zbl 0284.55016)]. The purpose of this paper is to construct a category \({\mathcal G}\) with the classifying space \(B{\mathcal G}\) having the homotopy type of \(\Omega^\infty \Sigma^\infty\) (Section 2) and to give two related results (Sections 3-6). The number of pages devoted to the first subject is rather small. But the key idea of the author seems to be a new concept “pseudo-tangle” introduced to construct \({\mathcal G}\).
This can be sketched as follows. The object of \({\mathcal G}\) is defined to be a word \(w\) consisting of a finite number of letters with the sign\(+\) or \(-\). This is pictured on a line \({\mathfrak R}\) by representing its letters by dots together with their signs. Then a morphism from \(w_1\) to \(w_2\) is defined to be an unodered set of paths in \({\mathfrak R}[0,1]\) subject to the following conditions: Each path connects two different points of \(w_1\times\{0\} \cup w_2\times \{1\}\) and also each point is in the image of exactly one path. Moreover each path connects either a point of \(w_1\times \{0\}\) with a point \(w_2\times \{1\}\) having the same sign or two points of \(w_2\times \{1\}\) having different signs. The author call such a picture a “pseudo-tangle”. One can get two operations in such a pictorial manner as stacking pseudo-tangles and putting words and pseudo-tangles side by side. Then \({\mathcal G}\) becomes a permutative category with them as composition and tensor product required. Let \({\mathcal F}_p\) be the full permutative subcategory of \({\mathcal G}\) whose object is a word consisting of only letters with plus sign. By applying Quillen’s \(S^{-1}S\) construction to \({\mathcal F}_P\) one obtains a new category \({\mathcal F}_P^{-1}{\mathcal F}_P\) and a canonical functor \({\mathcal F}_P\to {\mathcal F}_P^{-1} {\mathcal F}_P\) which induces a group completion \(B{ \mathcal F}_P\to B({\mathcal F}_P^{-1} {\mathcal F}_P)\). Then the main theorem (Theorem 2.4) states that \({\mathcal G}\) is equivalent to \({\mathcal F}_P^{-1}{\mathcal F}_P\), so that the first statement mentioned above (Corollary 2.5) follows immediately because \(B{ \mathcal F}_P=\coprod_{n\geq 0}B\Sigma_n\). These results are applied to study \(\pi^S_n\) for \(n=0,1,2\) and the homotopy fibre of the inclusion \({\mathcal G}\to{ \mathcal F}_{P,D}\) where \({\mathcal F}_{P,D}\) denotes the free permutative category with duality. Indeed a large number of illustrations of pseudo-tangles are used in proving the theorems obtained there.

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
55Q45 Stable homotopy of spheres
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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[1] DOI: 10.1007/BFb0080003 · doi:10.1007/BFb0080003
[2] Karpilovsky G., London Mathematical Society Monographs New Series 2, in: The Schur Multiplier (1987)
[3] Kassel C., Graduate Texts in Mathematics 155, in: Quantum Groups (1994)
[4] DOI: 10.1016/0022-4049(80)90101-2 · Zbl 0447.18005 · doi:10.1016/0022-4049(80)90101-2
[5] Mac Lane S., Graduate Texts in Mathematics 5, in: Categories for the Working Mathematician (1971) · Zbl 0232.18001
[6] DOI: 10.1007/BFb0067053 · Zbl 0292.18004 · doi:10.1007/BFb0067053
[7] DOI: 10.1016/0040-9383(74)90022-6 · Zbl 0284.55016 · doi:10.1016/0040-9383(74)90022-6
[8] DOI: 10.1016/0022-4049(92)00039-T · Zbl 0803.18004 · doi:10.1016/0022-4049(92)00039-T
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