Intersection number of immersions
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1 Introduction
Let ,
be immersions of oriented
- and
-manifolds in a connected oriented manifold of dimension
. 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 be a connected oriented manifold of dimension
and
![\displaystyle I_M=\lambda_M=\lambda: H_{n_1}(M)\times H_{n_2}(M) \to \Z](/images/math/e/5/6/e5642378fce81a594f4b3cbafc870f97.png)
the homology intersection pairing (or product) of .
Let be oriented
-manifolds
.
The double point set of maps
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 immersions
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
have transverse intersection (or are transverse) 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.
For any transverse immersions ,
of oriented
- and
-manifolds
![\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 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