We also give necessary and sufficient conditions for moving three maps f_1,f_2,f_3:S^2 --> X^4 to a position in which they have disjoint images. Again the obstruction lambda(f_1,f_2,f_3) generalizes Wall's intersection number lambda(f_1,f_2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in dimension 3, our new invariant corresponds to the Milnor invariant mu(1,2,3), generalizing the Matsumoto triple to the non simply-connected setting.
Keywords. Intersection number, 4-manifold, Whitney disk, immersed 2-sphere, cubic form
AMS subject classification. Primary: 57N13. Secondary: 57N35.
Submitted: 6 August 2000. Published: 25 October 2000.
Rob Schneiderman, Peter Teichner
Dept. of Mathematics, University of California at Berkeley
Berkeley, CA 94720-3840, USA
Dept. of Mathematics, University of California at San Diego
La Jolla, CA 92093-0112, USA
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