## Algebraic & Geometric Topology

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

#### Abstract

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 1, Number 1 (2001), 1-29.

Dates
Accepted: 4 September 2000
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882581

Digital Object Identifier
doi:10.2140/agt.2001.1.1

Mathematical Reviews number (MathSciNet)
MR1790501

Zentralblatt MATH identifier
0964.57022

Subjects
Primary: 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]
Secondary: 57N35: Embeddings and immersions

#### Citation

Schneiderman, Rob; Teichner, Peter. Higher order intersection numbers of 2–spheres in 4–manifolds. Algebr. Geom. Topol. 1 (2001), no. 1, 1--29. doi:10.2140/agt.2001.1.1. https://projecteuclid.org/euclid.agt/1513882581

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