Homotopy type of a 2-category
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
The classical group completion theorem states that under a certain condition the homology of ΩBM is computed by inverting π0M in the homology of M. McDuff and Segal extended this theorem in terms of homology fibration. Recently, more general group completion theorem for simplicial spaces was developed. In this paper, we construct a symmetric monoidal 2-category A. The 1-morphisms of A are generated by three atomic 2-dimensional CW-complexes and the set of 2-morphisms is given by the group of path components of the space of homotopy equivalences of 1-morphisms. The main part of the paper is to compute the homotopy type of the group completion of the classifying space of A, which is shown to be homotopy equivalent to Z × BAut∞+.
Abstract
The classical group completion theorem states that under a certain condition the homology of ΩBM is computed by inverting π0M in the homology of M. McDuff and Segal extended this theorem in terms of homology fibration. Recently, more general group completion theorem for simplicial spaces was developed. In this paper, we construct a symmetric monoidal 2-category A. The 1-morphisms of A are generated by three atomic 2-dimensional CW-complexes and the set of 2-morphisms is given by the group of path components of the space of homotopy equivalences of 1-morphisms. The main part of the paper is to compute the homotopy type of the group completion of the classifying space of A, which is shown to be homotopy equivalent to Z × BAut∞+.
참고문헌 (10)
- S, Galatius, "Stable homology of automorphism groups of free groups, to appear in Ann. of Math", ~
- A. Hatcher, "Stabilization for the automorphisms of free groups with boundaries", Geom. Topol., 9, pp.1295~1336 (2005)
- C. Jensen, "Automorphisms of free groups with boundaries", Algebr. Geom. Topol., 4, pp.543~569 (2004)
- J. P. May, "E∞-Spaces, group completions and permutative categories", Lon-don Math. Soc. Lecture Note Ser., 11, pp.61~95 (1974)
- J. P. May, "Classifying space and fibrations", Mem. Amer. Math. Soc., 1(1)통권 : 155, pp.13~98 (1975)
- D. McDuff, "Configuration spaces of positive and negative particles", Topol-ogy, 14, pp.91~107 (1975)
- D. McDuff, "Homology fibrations and the group completion theorem", Invent. Math., 31, pp.279~284 (1976)
- W. Pitsch, "Homology fibrations and group completion, re-visited", Homology, Homotopy Appl., 6(1), pp.153~166 (2004)
- Y. Song, "Braid groups, mapping class groups and cate-gorical delooping", Math. Ann., pp.377~393 (2007)
- U. Tillmann, "On the homotopy of the stable mapping class groups", Invent. Math., 130, pp.257~275 (1997)