\(Pin\) 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.

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 |