Volume 1, issue 1 (2001)

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Higher order intersection numbers of 2–spheres in 4–manifolds

Rob Schneiderman and Peter Teichner

Algebraic & Geometric Topology 1 (2001) 1–29

arXiv: math.GT/0008048


This is the beginning of an obstruction theory for deciding whether a map f : S2 X4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall’s self-intersection number μ(f) which tells the whole story in higher dimensions. Our second order obstruction τ(f) is defined if μ(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of π1X modulo S3–symmetry (rather then just one copy modulo S2–symmetry). It generalizes to the non-simply connected setting the Kervaire–Milnor invariant which corresponds to the Arf–invariant of knots in 3–space.

We also give necessary and sufficient conditions for moving three maps f1,f2,f3: S2 X4 to a position in which they have disjoint images. Again the obstruction λ(f1,f2,f3) generalizes Wall’s intersection number λ(f1,f2) 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 μ(1,2,3), generalizing the Matsumoto triple to the non simply-connected setting.

intersection number, 4–manifold, Whitney disk, immersed 2–sphere, cubic form
Mathematical Subject Classification 2000
Primary: 57N13
Secondary: 57N35
Forward citations
Received: 6 August 2000
Accepted: 4 September 2000
Published: 25 October 2000
Rob Schneiderman
Dept of Mathematics
University of California at Berkeley
Berkeley CA 94720-3840
Peter Teichner
Dept of Mathematics
University of California at San Diego
La Jolla CA 92093-0112