Jürgen Eckhoff, Ascona 1999
Jürgen Eckhoff is a German mathematician working in the areas of convexity and combinatorics. Our mathematical paths have met a remarkable number of times. We also met quite a few times in person since our first meeting in Oberwolfach in 1982. Here is a description of my mathematical dialogue with Jürgen Eckhoff:
Summary 1) (1980) we found independently two proofs for the same conjecture; 2) (1982) I solved Eckhoff’s Conjecture; 3) Jurgen (1988) solved my conjecture; 4) We made the same conjecture (around 1990) that Andy Frohmader solved in 2007, and finally 5) (Around 2007) We both found (I with Roy Meshulam, and Jürgen with Klaus Peter Nischke) extensions to Amenta’s Helly type theorems that both imply a topological version.
(A 2009 KTH lecture based on this post or vice versa is announced here.)
Let me start from the end:
5. 2007 – Eckhoff and I both find related extensions to Amenta’s theorem.
Nina Amenta proved a remarkable extension of Helly’s theorem. Let be a finite family with the following property:
(a) Every member of is the union of at most r pairwise disjoint compact convex sets.
(b) So is every intersection of members of .
If every r(d+1) members of has a point in common, then all members of have a point in common!
The case r=1 is Helly’s theorem, Grünbaum and Motzkin proposed this theorem as a conjecture and proved the case r=2. David Larman proved the case r=3.
Roy Meshulam and I studied a topological version of the theorem, namely you assume that every member of F is the union of at most r pairwise disjoint contractible compact sets in $R^d$ and that all these sets together form a good cover – every nonempty intersection is either empty or contractible. And we were able to prove it!
Eckhoff and Klaus Peter Nischke looked for a purely combinatorial version of Amenta’s theorem which is given by the old proofs (for r=2,3) but not by Amenta’s proof. An approach towards such a proof was already proposed by Morris in 1968, but it was not clear how to complete Morris’s work. Eckhoff and Nischke were able to do it! And this also implied the topological version for good covers.
The full results of Eckhoff and Nischke and of Roy and me are independent. Roy and I showed that if the nerve of is d-Leray then the nerve of is ((d+1)r-1)-Leray. Eckhoff and showed that if the nerve of has Helly number d, then the nerve of has Helly number (d+1)r-1. Amenta’s argument can be used to show that if the nerve of is d-collapsible then the nerve of F is ((d+1)r-1)-collapsible.
Here, a simplicial comples K is d-Leray if all homology groups vanishes for every and every induced subcomplex L of K.
Roy and I were thinking about a common homological generalization which will include both results but so far could not prove it.