fea-eckmann.qxp 3/5/99 3:45 PM Page 520 Topology, Algebra, Analysis—Relations and Missing Links Beno Eckmann his is largely, but not entirely, a histor- The second part is about the homotopy of the ical survey. It puts various matters to- unitary groups. It leads to a different linearization gether that are usually considered in phenomenon stating that maps of a sphere into a separate contexts. Moreover, it leads to unitary group are homotopic in the infinite unitary open, probably quite difficult problems group to a linear map (in the situation of the first Tand has analogues in contemporary mathematics. part it is, in general, not true that any continuous There are three parts. solution is homotopic to a linear one). Again, no di- The first part is about a certain range of clas- rect proof is known—by differential geometry, by sical results in algebraic topology concerning con- approximation, or by optimization. This lin- tinuous real functions and maps, vector fields, earization result is due to the close relation be- etc., that can be stated in a very simple way: just tween the classical Hurwitz-Radon matrices and replace “continuous” by “linear”. They thus seem Bott periodicity. Since the latter is the source of to be reduced to problems of algebra, essentially topological K-theory and general homology theo- linear algebra, where the solution is relatively easy. ries, a direct approach would establish a link be- The proofs of all these statements, however, do not tween an important old matrix problem and refined use such a reduction principle. They are beautiful methods of algebraic topology. and in general quite difficult, using elaborate ideas The third part is a short outlook concerning top- and techniques of topology such as cohomology ics of contemporary interest. It is about recent operations, spectral sequences, K-theory, and so trends in homology theories for spaces with infi- on. nite fundamental group G using Hilbert space In these cases the absence of a reduction prin- methods (Hilbert-G-modules, `2-homology, and ciple from continuity to linearity is thus a missing `2-Betti numbers). The presentation, necessarily a link between two areas. Of course, such a link is little technical, turns around the von Neumann al- not necessary since the results are proved. Still, it gebra of G considered just as an algebra, forget- might be interesting to have, at least in special ting the analysis behind it. We first recall the (weak) cases, a direct reduction to linearity which could Bass conjecture (1976), which has become a theo- throw new light onto old and new mathematics. rem for several classes of groups but is still open We begin, in that first part, with a very ele- in general, and give it a simple von Neumann al- mentary example. It contains in a nutshell the gebra formulation. Application to algebraic topol- problem to be discussed later for analogous but ogy and group homology again lead, in a very spe- more complex phenomena. cial example, to linearity. Beno Eckmann is professor emeritus of mathematics and We probably all agree that eventually reducing retired founding director of the Institute for Mathemati- a difficult problem to a “nice” situation is at the cal Research at Eidgenössische Technische Hochschule heart of mathematics. What I present here is a (ETH) in Zürich. modest attempt to list topics where a direct link 520 NOTICES OF THE AMS VOLUME 46, NUMBER 5 fea-eckmann.qxp 3/5/99 3:45 PM Page 521 to linearity or to easy algebra probably exists but circle determined by x and v(x) and move the is not known. If found, it might, even in different point x, in the v(x)-direction, to its antipode −x, contexts, have some general significance. But in for all x ∈ Sn−1. This is a homotopy (a continuous most other areas, nice situations certainly are of deformation) between the identity map and the an- a quite different nature; there is no need to list ex- tipodal map of Sn−1. amples. At this point the homological concept of degree comes into play. Its value is 1 for the identity and Vector Fields and Vector Functions (−1)n for the antipodal map. Homotopic maps Two preliminary remarks seem appropriate. have the same degree, so we get 1=(−1)n; i.e., n 1) The results in the first part are all about cer- is even. tain positive integers (dimensions, number of vec- tor fields, etc.). They state that, apart from those Proposition 2. A continuous tangent unit vector n−1 − values of these integers where (multi-)linear solu- field on S exists if and only if n 1 is odd. tions exist, there are no continuous solutions. This The crucial point is now to express these facts negative statement can be turned into a positive in a different way. We write Pn for the problem: one, mainly about the existence of zeros; see Propo- Does there exist on Sn−1 a tangent unit vector sition 3. Moreover, in some of those cases where field? a linear solution exists, there are also other con- tinuous solutions of a different nature (not ho- Theorem L (“Linearization”). If Pn has a continu- motopic to linear ones). ous solution, then it also has a linear solution. 2) We, of course, do not suggest that the re- Theorem L is proved “indirectly” by using a spective topological results should have been method from algebraic topology. There is no harm proved from the outset by reduction to linearity. in doing so. By a “direct” proof we would mean a On the contrary, these problems were a source of procedure replacing a continuous field by a linear stimulation for developing interesting tools that one (for example, through a variational principle still are very important, e.g., in modern homotopy where in the space of all continuous fields there theory. The linearization phenomenon emerged would be an extremal expected to be linear). Such only a posteriori after the topological theorems had a direct proof would reduce the topological prob- been established. lem to very elementary linear algebra. Tangent Vector Field on a Sphere In the more complicated situations to be de- n−1 n A tangent unit vector field v on S P⊂ R (given scribed below, a direct reduction to a transparent Rn 2 in coordinates ξ1,... ,ξn of by ξi =1) is a algebraic argument, though not necessary, might n−1 function that attaches to x ∈ S a vector v(x) sat- be even more interesting. isfying Remarks. 1) The negative statement “There is, for h i (1) v(x),x =0, even n − 1, no continuous tangent unit vector field n−1 (2) |v(x)|2 = hv(x),v(x)i =1 on S ” can be turned into a nontrivial existence statement for zeros, as follows. for all x ∈ Sn−1. Here h , i is the standard scalar product in Rn. Proposition 3. Let fi(ξ1,... ,ξn), i =1,... ,n, be continuous functions satisfying Linearity implies that (2) is equivalent to X 0 ξ f (ξ ,... ,ξ )=0 (2 ) |v(x)|2 = hx, xi for all x ∈ Rn. i i 1 n If n is even, v(x)=(−ξ2,ξ1, −ξ4,ξ3,...) is such for all x with |x| =1. If n is odd, then the fi have a field, linear in x. What about linear fields if n is a common zero. odd? Let X Otherwise the f could be normalized so as to v = a ξ i i ik k be the components of a unit tangent vector field on Sn−1. If the f are polynomials, one has an al- be the components of v(x). Then (1) means i P gebraic statement for which no algebraic proof a ξ ξ =0for all x; i.e., the matrix a is skew- ik i k ik seems to be known. For the complex analogue, symmetric. If n is odd, its determinant is 0, and however, there is an algebraic proof by van der thus there is an x ∈ Sn−1 with v(x)=0, in contra- Waerden (1954). diction to (2). 2) The above proof of Proposition 2 is by ele- Proposition 1. A linear tangent unit vector field − mentary algebraic topology. For differentiable vec- on Sn 1 exists if and only if n − 1 is odd. tor fields there are other classical proofs, using We now ask the same question for continuous analysis or geometry; they are all based on some vector fields. These are, of course, more interest- version of the concept of degree. A very different ing from the viewpoint of geometry and analysis. analytic proof, however, is due to Milnor (1978). Given such a field on Sn−1, we consider the great None of these proofs is by reduction to linearity. MAY 1999 NOTICES OF THE AMS 521 fea-eckmann.qxp 3/5/99 3:45 PM Page 522 Vector Functions of Two (or More) Variables, Because of the “composition of quadratic forms” Multiplications in Rn given by (7), the algebras with norm product rule The well-known vector cross-product x × y in R3 are also called composition algebras. In 1898 Hur- is a function of two vectors that is bilinear and ful- witz proved that such a “composition of quadratic fills forms” with bilinear functions j of the ξj and ηj, with real or complex coefficients, can exist for h × i h × i (3) x y,x = x y,y =0, n =1, 2, 4, 8 only. | × |2 | |2| |2 −h i2 (4) x y = x y x, y .