Intersection number of immersions
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1 Introduction
This page is based on [Ranicki2002]. Let ,
be immersions of oriented manifolds in a connected oriented manifold. The intersection number
has both an algebraic and geometric formulation; roughly speaking it 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. The intersection number of immersions is closely related to the intersection form of a
-dimensional manifold and in turn its signature: important invariants used in the classification of manifolds.
2 Definition
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![m](/images/math/f/5/2/f52ba22baf75438bb1b02f476954c023.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle \lambda: H_n(M)\times H_{m-n}(M) \to \Z; \quad (x,y) \mapsto \lambda(x,y),](/images/math/2/a/7/2a779273304a99a6186c4c0c71431a5a.png)
![\displaystyle \lambda(x,y) = \langle x^*\cup y^*,[M]\rangle \in \Z](/images/math/1/e/c/1ecb41c6f078d2b691a5bbfe5e2d025d.png)
![x^*\in H^{m-n}(M)](/images/math/3/8/9/389638f7c9e68502c4cb12c0edaae7eb.png)
![y^*\in H^n(M)](/images/math/2/f/f/2ff51a414a78dbb93cb4c968d52f217a.png)
![x](/images/math/8/7/2/8725029ea89712eed8670bae64d30e47.png)
![y](/images/math/3/6/a/36a4dc9ccf2bdc09d800556724231fc6.png)
![[M]](/images/math/f/a/0/fa08c3d5d2f54260952acc8a646b5025.png)
As a consequence of the properties of the cup product the homology intersection pairing is bilinear and satisfies
![\displaystyle \lambda(y,x) = (-1)^{n(m-n)}\lambda(x,y)](/images/math/f/f/f/fffd7d37c906d51f562368edbd4023a5.png)
for all ,
.
![f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}](/images/math/a/c/c/accca7e6e9770caaf94d1debbb012913.png)
![f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}](/images/math/c/7/a/c7a77f2159e48cc7d1b3b703821e7819.png)
![\lambda^{\mathrm{alg}}(N_1,N_2)\in\Z](/images/math/a/f/8/af89a109becf380387c97ab4f39ce06c.png)
![(f_1)_*[N_1]\in H_{n_1}(M)](/images/math/5/0/9/509ee0909398b7f1fdcd3ac61033f43e.png)
![(f_2)_*[N_2]\in H_{n_2}(M)](/images/math/6/5/2/6526c09939b70b354f1e605991ddea1b.png)
![\displaystyle \lambda^{\mathrm{alg}}(N_1,N_2) := \lambda((f_1)_*[N_1],(f_2)_*[N_2]).](/images/math/4/b/8/4b8c399582236de311aa3eb060ed9393.png)
3 Alternative description
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/3/8/5/38536fb446103fc62d708d0d9b66089f.png)
with the diagonal subspace.
![x=(x_1,x_2)\in S_2(f_1,f_2)](/images/math/4/7/f/47fddbdc22a7d03c5eb73d89ef99e66f.png)
![f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}](/images/math/5/0/4/504ba71c9683bbe6d67ec80836b2413d.png)
![(i=1,2)](/images/math/a/5/a/a5aac2230cb3046efaae7252f3efaa09.png)
![\displaystyle df(x) = (df_1(x_1),df_2(x_2)): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_M(f(x))](/images/math/d/0/0/d00bb668682120dd0f1608d859ca27ff.png)
Immersions
have transverse intersections (or are transverse) if each double point is transverse and
is finite.
![I(x)\in\Z](/images/math/8/8/0/880876cd5d944538d355c20bbfa93f42.png)
![x=(x_1,x_2)\in S_2(f_1,f_2)](/images/math/4/7/f/47fddbdc22a7d03c5eb73d89ef99e66f.png)
![\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)
The geometric intersection number of transverse immersions
is
![\displaystyle \lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z.](/images/math/8/0/a/80a6215d3a803465c84178fca36b366c.png)
4 Equivalence of definitions
![\displaystyle \lambda^{\mathrm{alg}}(N_1,N_2)=\lambda^{\mathrm{geo}}(N_1,N_2).](/images/math/b/d/1/bd15a416840ea05a4ed6dcef61c0518f.png)
5 References
- [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