## $$Pin$$ structures on low-dimensional manifolds.(English)Zbl 0754.57020

Geometry of low-dimensional manifolds. 2: Symplectic manifolds and Jones- Witten-Theory, Proc. Symp., Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 151, 177-242 (1990).
[For the entire collection see Zbl 0722.00024.]
The orthogonal group $$O(n)$$ can be introduced starting with a definite quadratic form on $$\mathbb{R}^ n$$ irrespective of the fact whether this form is positive or negative. But when we construct a covering of $$O(n)$$ via the Clifford algebras, the choice of sign is relevant, and we obtain two different central extensions $$Pin^ +(n)$$ and $$Pin^ -(n)$$ of $$O(n)$$. (They are homeomorphic but not isomorphic as groups). The covering $$Spin(n)\to SO(n)$$ enables to introduce the notion of $$Spin$$ structure on an oriented $$n$$-dimensional vector bundle, and similarly the notion of $$Pin^ \pm$$ structure on a non-oriented $$n$$-dimensional vector bundle can be introduced.
In the first section the authors investigate relations among $$Spin$$, $$Pin^ +$$, and $$Pin^ -$$ structures on various vector bundles (e.g., there is a bijection between $$Pin^ -$$ structures on a vector bundle $$\xi$$ and $$Spin$$ structures on $$\xi\oplus \text{det}\xi$$), and show that a $$Pin^ \pm$$ structure on the tangent bundle of a manifold descends to a $$Pin^ \pm$$ structure on a codimension-one submanifold with a trivialized normal bundle. This enables to introduce $$Pin^ \pm$$ bordism groups $$\Omega_ m^{Pin^ \pm}$$.
In the second section the authors show that $$\Omega^{Spin}_ 0\cong\mathbb{Z}$$, $$\Omega^{Pin^ \pm}_ 0\cong\mathbb{Z}/2\mathbb{Z}$$, $$\Omega_ 1^{Spin}\cong\mathbb{Z}/2\mathbb{Z}$$, $$\Omega^{Pin^ +}_ 1\cong 0$$, $$\Omega^{Pin^ -}_ 1\cong\mathbb{Z}/2\mathbb{Z}$$. They prove here also another kind of descent theorem which puts a $$Pin^ \pm$$ structure on a submanifold dual to a characteristic class. (E.g., they obtain a homomorphism $$[\cap w^ 2_ 1]:\Omega_ m^{Pin^ \pm}\to\Omega^{Pin^ \mp}_{m-2}$$.)
The third section is devoted to $$Pin^ \pm$$ structures on surfaces. It is shown here that there is a canonical bijection between the $$Pin^ -$$ structures on a surface and the quadratic enhancements of its intersection form. Then, using the fact that the Brown invariant is an invariant of the $$Pin^ -$$ bordism, the authors prove that $$\Omega_ 2^{Pin^ -}\cong\mathbb{Z}/8\mathbb{Z}$$. They prove also that $$\Omega^{Pin^ +}_ 2\cong \Omega^{Spin}_ 1\cong\mathbb{Z}/2\mathbb{Z}$$.
The fourth section deals with 3-dimensional $$Spin$$ manifolds. The authors present here a geometric interpretation of the Turaev’s results on the trilinear intersection forms. Using this, they prove the following relation: Let $$\Theta_ 1$$ and $$\Theta_ 2$$ be two Spin structures on a 3-dimensional manifold $$M$$, and let $$\alpha\in H^ 1(M;\mathbb{Z}/2\mathbb{Z})$$ be the element which acting on $$\Theta_ 1$$ gives $$\Theta_ 2$$. Then for the $$\mu$$-invariants of both $$Spin$$ structures we have $$\mu(\Theta_ 2)=\mu(\Theta_ 1)-2\beta(a){}\bmod 16$$, where $$a\in H_ 2(M;\mathbb{Z}/2\mathbb{Z})$$ is the Poincaré dual to $$\alpha$$.
In the fifth section the authors compute $$\Omega^{Spin}_ 3\cong 0$$, $$\Omega^{Pin^ -}_ 3\cong 0$$, $$\Omega^{Pin^ +}_ 3\cong\Omega^{Spin}_ 2\cong\mathbb{Z}/2\mathbb{Z}$$, $$\Omega^{Spin}_ 4\cong\mathbb{Z}$$, $$\Omega^{Pin^ -}_ 4\cong 0$$, $$\Omega^{Pin^ +}_ 4\cong\mathbb{Z}/16\mathbb{Z}$$.
The sixth section starts with 4-dimensional manifolds $$M$$ and surfaces $$F\subset M$$ dual to $$w_ 2+w^ 2_ 1$$. A pair $$(M,F)$$ is called characterized if there is a $$Pin^ -$$ structure on $$M-F$$ which does not extend across any component of $$F$$. We find here a generalization of the Guillou-Marin formula. Then the authors pass to a manifold $$M$$ of arbitrary dimension $$r$$ and a submanifold $$F\subset M$$ of codimension two. They introduce a notion of characteristic structure on $$(M,F)$$, and show that it is possible to define the bordism group $$\Omega^ !_ r$$ of characteristic structures.
In the seventh section the authors compute $$\Omega^ !_ 0\cong\Omega^ !_ 2\cong\mathbb{Z}/2\mathbb{Z}$$, $$\Omega^ !_ 1\cong 0$$, $$\Omega^ !_ 3\cong\Omega^{Pin^ -}_ 1\cong\mathbb{Z}/2\mathbb{Z}$$, $$\Omega^ !_ 4\cong\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$$.
In the eighth section, being inspired by the Robertello’s approach to the Arf invariant of a knot, the authors use $$\beta: \Omega^{Pin^ -}_ 2\to \mathbb{Z}/8\mathbb{Z}$$ to introduce a $$\mathbb{Z}/8\mathbb{Z}$$-invariant to a characteristic link $$L$$ in a $$Spin$$ 3-dimensional manifold $$M$$ with a given set of even longitudes.
The last, ninth section contains comments on the 4-dimensional topological manifold versions of some of the 4-dimensional differential manifold results.
Reviewer: J.Vanžura (Brno)

### MSC:

 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) 57N10 Topology of general $$3$$-manifolds (MSC2010) 57R90 Other types of cobordism 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57R20 Characteristic classes and numbers in differential topology

Zbl 0722.00024