Intersection number of immersions
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1 Introduction
Let be a connected oriented manifold of dimension
and
,
immersions of oriented
- and
-manifolds. The intersection number of
and
counts with sign the number of intersection points that the two immersions have. The intersection number is an obstruction to perturbing the immersions into being disjoint. When it vanishes this perturbation can often be achieved using the Whitney trick.
This page is based on [Ranicki2002], see also [Broecker&Jaenich1982, Excercise 14.9.6].
2 Statement
Let
![\displaystyle I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z](/images/math/e/5/6/e5642378fce81a594f4b3cbafc870f97.png)
be the homology intersection pairing (or product) of .
The double point set of and
is defined by
![\displaystyle S_2(f_1,f_2) = \{(x_1,x_2)\in N_1\times N_2 | f_1(x_1) = f_2(x_2)\in M\} = (f_1\times f_2)^{-1}(\Delta(M)),](/images/math/2/5/c/25cbc0cb2db0ac02d165945851e9c781.png)
where the diagonal.
A double point of
and
is transverse if the linear map
![\displaystyle df(x) = df_1(x_1)\oplus df_2(x_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))](/images/math/d/6/6/d66c1a9d412cda3088a4a696ac1fd03e.png)
is an isomorphism.
Immersions and
are transverse (or have transverse intersection) if
is finite and every double point is transverse.
The index, or the sign of a transverse double point
is
![\displaystyle I(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; df(x)\; \mathrm{preserves}\; \mathrm{orientations}\\ -1, & \mathrm{otherwise}.\end{array}\right.](/images/math/6/f/3/6f386805f2f2212e008fc846139715f3.png)
Theorem 2.1. If and
are transverse, then
![\displaystyle \lambda(f_{1*}[N_1],f_{2*}[N_2])=\sum_{x\in S_2(f_1,f_2)}{I(x)}.](/images/math/6/a/d/6adfe6a8168c927468d1cb8f9c5c707d.png)
This clasical fact is either a theorem or a definition depending on which definition of homology intersection pairing one accepts. For a proof see [Scorpan2005, Section 3.2] or [Ranicki2002, Proposition 7.22]. Unless this equality is a definition, the left- and right- hand sides of the equality can be called algebraic and geometric intersection number of and
.
References
- [Broecker&Jaenich1982] Th. Br\"ocker and K. J\"anich Introduction to Differential Topology, Cambridge University Press, 1982. ISBN-13: 978-0521284707, ISBN-10: 0521284708.
- [Ranicki2002] A. Ranicki, Algebraic and geometric surgery, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001
- [Scorpan2005] A. Scorpan, The wild world of 4-manifolds, American Mathematical Society, 2005. MR2136212 (2006h:57018) Zbl 1075.57001