#### Algebraic and Geometric Topology 1 (2001),
paper no. 1, pages 1-29.

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

### Rob Schneiderman, Peter Teichner

**Abstract**.
This is the beginning of an obstruction theory for deciding whether a
map f:S^2 --> X^4 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
mu(f) which tells the whole story in higher dimensions. Our second
order obstruction tau(f) is defined if mu(f) vanishes and has formally
very similar properties, except that it lies in a quotient of the
group ring of two copies of pi_1(X) modulo S_3-symmetry (rather then
just one copy modulo S_3-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
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.

**DOI:** 10.2140/agt.2001.1.1

**E-print:** `arXiv:math.GT/0008048`

Submitted: 6 August 2000.
Published: 25 October 2000.

Notes on file formats
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

Email: schneido@math.berkeley.edu, teichner@euclid.ucsd.edu

AGT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**