0000000000001572
AUTHOR
Peter Teichner
On the signature of four-manifolds with universal covering spin
In this note we study closed oriented 4-manifolds whose universal covering is spin and ask whether there are restrictions on the divisibility of the signature. Since any natural number appears as the signature of a connected sum of r 2,s, without the assumption on the universal covering there cannot exist any restrictions. Certainly, the most famous such restriction was proved by Rohlin in [10], where he showed that the signature a of a smooth 4-dimensional spin manifold is divisible by 16 (compare part (2) of our Main Theorem for a new proof). The Kummer surface K shows that this is the best possible general result. Dividing by a certain free holomorphic involution on K, one obtains the En…
4-Manifold topology I: Subexponential groups
The technical lemma underlying the 5-dimensional topological s-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating ~-null immersions of disks. These conjectures are theorems precisely for those fundamental groups ("good groups") where the ~l-null disk lemma (NDL) holds. We expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two opera- tions: (1) extension and (2) direct limit. The finitely generated groups in this class are amenable and no amenable group is known to lie outside this class.
4-Manifold topology II: Dwyer's filtration and surgery kernels
Even when the fundamental group is intractable (i.e. not "good") many interesting 4-dimensional surgery problems have topological solutions. We unify and extend the known examples and show how they compare to the (presumed) counterexamples by reference to Dwyer's filtration on second homology. The development brings together many basic results on the nilpotent theory of links. As a special case, a class of links only slightly smaller than "homotopically trivial links" is shown to have (free) slices on their Whitehead doubles.
Counterexamples to the Kneser conjecture in dimension four.
We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM 0#M 1 unlessM 0 orM 1 is homeomorphic toS 4. LetN be the nucleus of the minimal elliptic Enrique surfaceV 1(2, 2) and putM=N∪ ∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S 2×S2) is diffeomorphic toM 0#M 1 for non-simply connected closed smooth four-manifoldsM 0 andM 1 if and only ifk≥8. On the other hand we show thatM is homeomorphic toM 0#M 1 for closed topological four-manifoldsM 0 andM 1 withπ 1(Mi)=ℤ/2.