Equivariant loops on classifying spaces

Moi, Kristian Jonsson¹

¹Department of Mathematics, KTH, Stockholm, Sweden

Algebraic and Geometric Topology, Tome 20 (2020) no. 5, pp. 2511-2552

Résumé

We compute the homology of the space of equivariant loops on the classifying space of a simplicial monoid M with anti-involution, provided π0(M) is central in the homology ring of M. The proof is similar to McDuff and Segal’s proof of the group completion theorem. Then we give an analogous computation of the homology of the C2–fixed points of a Γ–space-type delooping of an additive category with duality with respect to the sign circle. As an application we show that this fixed-point space is sometimes group complete, but in general not.

Détail

DOI : 10.2140/agt.2020.20.2511

Classification : 55P35, 55P48, 55P91
Keywords: algebraic topology, loop space, classifying space, group completion

Bibliographie

[1] E Dotto, Stable real K–theory and real topological Hochschild homology, PhD thesis, University of Copenhagen (2012)

[2] P G Goerss, J F Jardine, Simplicial homotopy theory, 174, Birkhäuser (1999) | DOI

[3] D Grayson, Higher algebraic K–theory, II (after Daniel Quillen), from: "Algebraic –theory" (editor M R Stein), Lecture Notes in Math. 551, Springer (1976) 217 | DOI

[4] L Hesselholt, I Madsen, Real algebraic K–theory, preprint (2015)

[5] J F Jardine, Lectures on algebraic K–theory, lecture notes (2014)

[6] M Karoubi, O Villamayor, K–théorie algébrique et K–théorie topologique, II, Math. Scand. 32 (1973) 57 | DOI

[7] S Mac Lane, Categories for the working mathematician, 5, Springer (1998) | DOI

[8] D Mcduff, G Segal, Homology fibrations and the “group-completion” theorem, Invent. Math. 31 (1976) 279 | DOI

[9] J Milnor, D Husemoller, Symmetric bilinear forms, 73, Springer (1973) | DOI

[10] W Pitsch, J Scherer, Homology fibrations and “group-completion” revisited, Homology Homotopy Appl. 6 (2004) 153 | DOI

[11] D Quillen, On the group completion of a simplicial monoid | DOI

[12] M Schlichting, Hermitian K–theory on a theorem of Giffen, –Theory 32 (2004) 253 | DOI

[13] M Schlichting, Hermitian K–theory of exact categories, J. –Theory 5 (2010) 105 | DOI

[14] M Schlichting, The Mayer–Vietoris principle for Grothendieck–Witt groups of schemes, Invent. Math. 179 (2010) 349 | DOI

[15] G Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973) 213 | DOI

[16] G Segal, Categories and cohomology theories, Topology 13 (1974) 293 | DOI

[17] K Shimakawa, Infinite loop G–spaces associated to monoidal G–graded categories, Publ. Res. Inst. Math. Sci. 25 (1989) 239 | DOI

[18] N A Stiennon, The moduli space of real curves and a Z∕2–equivariant Madsen–Weiss theorem, PhD thesis, Stanford University (2013)

[19] R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91 | DOI

[20] F Waldhausen, Algebraic K–theory of spaces, from: "Algebraic and geometric topology" (editors A Ranicki, N Levitt, F Quinn), Lecture Notes in Math. 1126, Springer (1985) 318 | DOI

[21] C T C Wall, On the axiomatic foundations of the theory of Hermitian forms, Proc. Cambridge Philos. Soc. 67 (1970) 243 | DOI