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

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

### Citations:

Zbl 0195.249; Zbl 0167.216; Zbl 0268.57016
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