Kervaire-Milnor invariant

An invariant of framed surgery of a closed 6- or 14-dimensional framed manifold.

Let $ M ^ {6} $ be a stably-parallelizable 2-connected manifold on which is given a stable $ N $-dimensional framing $ ( M ^ {6} , U ) $, i.e. a trivialization of the stable $ N $-dimensional normal bundle. Let $ S _ {i} ^ {3} $ be spheres realizing a basis of the 3-dimensional homology space of $ M ^ {6} $. By summing the given $ N $-trivialization $ U $ with certain trivializations $ \alpha _ {i} \in \pi _ {3} ( \mathop{\rm SO} _ {3} ) $ of tubular neighbourhoods of the spheres $ S _ {i} ^ {3} $ in $ M ^ {6} $, one obtains an $ ( N + 3 ) $-dimensional trivialization of the stable normal bundles to the spheres $ S _ {i} ^ {3} $ and the corresponding elements $ \alpha _ {i} ^ {1} \in \pi _ {3} ( \mathop{\rm SO} _ {N+ 3} ) $. The cokernel of the stable homomorphism $ s : \pi _ {n} ( \mathop{\rm SO} _ {N+ 3} ) \rightarrow \pi _ {n} ( \mathop{\rm SO} _ {N+ n} ) $ is isomorphic to $ \mathbf Z _ {2} $ for $ n = 3 $, so that each sphere $ S _ {i} ^ {3} $ is put into correspondence with an element of the group $ \pi _ {3} ( \mathop{\rm SO} _ {N+ 3} ) / \mathop{\rm Im} s $ (according to the value of the elements $ \alpha _ {i} ^ {1} $ which they take in the group $ \mathbf Z _ {2} $ after factorization by $ \alpha _ {i} ^ {1} $). This value does not depend on the choice of the elements $ \alpha _ {i} $, but depends only on the homology classes realized by the spheres $ S ^ {3} $ and the framing $ U $. The Arf-invariant of the function $ \phi _ {0} : H _ {3} ( M ^ {6} , \mathbf Z ) \rightarrow \mathbf Z _ {2} $ so obtained satisfies the formula $ \phi _ {0} ( x + y ) = \phi _ {0} ( x) + \phi _ {0} ( y) + \phi ( x, y ) $ $ \mathop{\rm mod} 2 $, where $ \phi ( x , y ) $ is the intersection form of the 3-dimensional homology space on the manifold $ M ^ {6} $, and is called the Kervaire–Milnor invariant of this manifold with framing $ U $. The pair $ ( M ^ {6} , U ) $ has a framed surgery to the pair $ ( S ^ {6} , V ) $ if and only if the Kervaire–Milnor invariant of $ ( M ^ {6} , U ) $ is zero.

Similar constructions have been carried out for $ M ^ {14} $. The Kervaire–Milnor invariant in dimension six is the only invariant of the stable 6-dimensional framed cobordism and defines an isomorphism $ \pi _ {n+ 6} ( S ^ {n} ) \approx \mathbf Z _ {2} $, $ n \geq 7 $. However, in dimension fourteen it is not a unique invariant of the stable 14-dimensional framed cobordism, i.e. the stable group $ \pi _ {n+14} ( S ^ {n} ) $, $ n \geq 16 $, is defined by framings on the sphere $ S ^ {14} $ and on $ S ^ {7} \times S ^ {7} $.

For references see Kervaire invariant.