A calculation of \(Pin^ +\) bordism groups. (English) Zbl 0719.55004

Kirby, R. C.; Taylor, L. R.

Comment. Math. Helv. 65, No. 3, 434-447 (1990).

The groups \(Pin^{\pm}(n)\) are non-trivial double covers of O(n); their classifying spaces B Pin\({}^{\pm}(n)\) are induced from the maps BO(n)\(\to K({\mathbb{Z}}/2,2)\) given by \(w_ 2+w^ 2_ 1\) for \(Pin^-(n)\) and by \(w_ 2\) for \(Pin^+(n)\). Let M Pin\({}^{\pm}\) denote the corresponding bordism theories of manifolds with Pin\({}^{\pm}\)- structure. There are well-known equivalences \(M Pin^{\pm}\simeq M Spin\wedge \Sigma^{-m}P_ m^{\infty}\) (Lemma 6) where \(P_ m^{\infty}\) is the stunted real projective space and \(m=4k+3\) for \(Pin^+\) and \(m=4k+1\) for \(Pin^-\). The (more classical) groups M Pin\({}^-_*\) were computed by D. W. Anderson, E. H. Braun jun. and F. P. Peterson [Comment. Math. Helv. 44, 462-468 (1969; Zbl 0195.249)] based on their analysis of Spin bordism. The first result of the present paper is the analogous computation for the case M Pin\({}^+_*\). (The actually needed computation of \(bo<n>_*P_ m^{\infty}\) seems to be first published by S. Gitler, M. Mahowald and R. J. Milgram [Proc. Natl. Acad. Sci. USA 60, 432-437 (1968; Zbl 0167.216)]). The second result is the more geometric investigation of the behaviour of the manifolds \({\mathbb{R}}P^{2n}\) in M Pin\({}_*^{\pm}\). This is based on the cofibrations \(P_ m^{\infty}\to P^{\infty}_{m+2}\) and corrects some statements of V. Giambalvo [Proc. Am. Math. Soc. 39, 395-401 (1973; Zbl 0268.57016)].

Reviewer: E.Ossa (Wuppertal)

MSC:

55N22	Bordism and cobordism theories and formal group laws in algebraic topology
57R90	Other types of cobordism

Keywords:

classifying spaces; bordism theories; \(Pin^{\pm }\)-structure

Citations:

Zbl 0195.249; Zbl 0167.216; Zbl 0268.57016