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2001 Higher order intersection numbers of 2–spheres in 4–manifolds
Rob Schneiderman, Peter Teichner
Algebr. Geom. Topol. 1(1): 1-29 (2001). DOI: 10.2140/agt.2001.1.1


This is the beginning of an obstruction theory for deciding whether a map f:S2X4 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:S2X4 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.


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Rob Schneiderman. Peter Teichner. "Higher order intersection numbers of 2–spheres in 4–manifolds." Algebr. Geom. Topol. 1 (1) 1 - 29, 2001.


Received: 6 August 2000; Accepted: 4 September 2000; Published: 2001
First available in Project Euclid: 21 December 2017

zbMATH: 0964.57022
MathSciNet: MR1790501
Digital Object Identifier: 10.2140/agt.2001.1.1

Primary: 57N13
Secondary: 57N35

Rights: Copyright © 2001 Mathematical Sciences Publishers


Vol.1 • No. 1 • 2001
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