On the Hopf Index Theorem and the Hopf Invariant
نویسنده
چکیده
Let ƒ: N —• M be a C°° map of oriented compact manifolds, and let L be an oriented closed submanifold of codimension q > 1 in M. If w is a closed form Poincaré dual to L, we show that f~L, with multiplicities counted, is Poincaré dual to ƒ *w in N and is even meaningful on a "secondary" level. This leads to generalized versions of the Hopf invariant, the Hopf index theorem and the Bezout theorem. We assume that the connected components Tl9 . . . , Tz of f~ L are submanifolds of codimension q in N. Let ord 1̂ . be the intersection number of L and ƒ \Bt where B is a qr-dimensional submanifold meeting r,. transversally at a single point. A proper choice of orientations makes ord T( > 0.
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