From: lrudolph@panix.com (Lee Rudolph) Subject: Re: 4-dim space .... Date: 22 Jul 1999 07:28:51 -0400 Newsgroups: sci.math Keywords: almost-complex structures Pertti Lounesto writes: >OK. If the rotation angles are the same, then every point on the unit >sphere is in some invariant plane, but not every plane is an invariant >plane. The invariant planes interesect the unit sphere S^3 of R^4 in >circles, which are pairwise linked in S^3. A pair of invariant planes, >which are orthogonal to each other, with fixed senses of rotations, >induces a pair of senses of rotations for all pairs of ortogonal planes. >Such sets of pairs of orthogonal planes can be divided into two >classes: they correspond to left and right multiplication, q -> aq and >q -> qb, when an arbitrary rotation of R^4 is written by means of >unit quaternions a,b as q -> aq/b. Here's an application, dear to my own heart, of what Pertti's discussing above. Recall that an _almost-complex structure_ on a real vectorspace V is a linear automorphism J of V which is a "square root of -1", that is, such that J^2 = -I (where I is the identity of V). [Exercise: if V has an almost complex structure and is finite dimensional, then its dimension is even. Example: let V be the real vectorspace underlying a complex vectorspace, and let J be the real linear automorphism underlying the complex linear automorphism "multiplication by i".] When V is not just a vectorspace, but is also equipped with a Euclidian structure, then we can define an _orthogonal almost-complex structure_ to be an almost-complex structure which is orthogonal (duh). Let V be the real vectorspace underlying the quaternions H, equipped with their usual Euclidian structure. Clearly, for any pure unit quaternion q = ai+bj+ck (a^2+b^2+c^2=1), both L(q) (left multiplication by q, L(q)v=qv) and R(q) (right multiplication by q, R(q)v=vq) is an orthogonal almost-complex structure on V. THEOREM. Every orthogonal almost-complex structure on V is either L(q) or R(q) for some pure unit quaternion q. Given any oriented real 2-plane P in V, there is one and only one pair (q,q') of pure unit quaternions such that (1) L(q)P = P = R(q')P and (2) the orthogonal almost-complex structures L(q)|P and R(q')|P on P respect the given orientation of P (that is, if v is a non-zero vector in P, then (v,qv) and (v,vq') are positively oriented bases of P). This (easy) theorem gives a nice, explicit bijection (which is, of course, a diffeomorphism) between the Grassmann manifold G of oriented 2-planes in V=R^4 and the product S^2 x S^2 of two unit 2-spheres (where S^2 is the 2-sphere of pure unit quaternions). The explicitness of the bijection means, for instance, that in any situation where you have a field of oriented 2-planes on a submanifold M of R^4, you get a pair of *explicit* maps from M to S^2, and so you have some hope of making *explicit* calculations of invariants of the field you started with. (Examples: M might be a surface, say a minimal surface, and you can look at the field of tangent planes. Or M might be a 3-manifold in R^4 equipped with an oriented foliation or contact structure, and you can look at the field of planes tangent to the foliation or contact structure. Or--the case I know best--M might be the punctured neighborhood in R^4 of an isolated critical point of a function F from R^4 to R^2, and you can look at the field of kernels of the differential of F. In this last case, you have--up to homotopy--a pair of maps from S^3 to S^2, and thus by passing to Hopf invariants a pair of integers which are intrinsically associated to the isolated critical point. If F is the real mapping underlying a complex polynomial from C^2 to C, then one of these integers is 0 and the other is the Milnor number of the critical pont. In other cases more interesting things happen. The explicitness I've been hymning means that, in *some* other cases anyway, you stand some hope of computing these two integers.) Lee Rudolph ============================================================================== From: Lee Rudolph Subject: Re: 4-dim space .... Date: Fri, 23 Jul 1999 06:57:52 -0400 (EDT) Newsgroups: [missing] To: rusin@math.niu.edu (Dave Rusin) > Nice post! Thanks for taking the time to describe this. > > I take it the "Milnor number" is an invariant described by him in his > book "Singular points of complex hypersurfaces" but it has been many > years since I really read that book. Can you remind me what that number is? He begins by defining it as the "multiplicity" of the zero of the (real) gradient vectorfield of the function f:C^n->C with the isolated critical point in question, that is, as the degree of the map from a tiny S^{2n-1} to S^{2n-1} obtained by restricting the vectorfield to the tiny sphere and normalizing it. He then very quickly shows that this integer is, in fact, a non-negative integer, by showing it's equal to the rank of the middle homology of the "Milnor fiber" of the isolated critical point (the (2n-2)-manifold-with-boundary, well defined up to isomorphism, which is the intersection of the tiny D^2n bounded by the tiny S^{2n-1} with f^{-1}(c), where c is a regular value of f which is very very close to the critical value at the isolated critical point in question). The latter definition has long been universally taken as the primary definition, since it generalizes in a much more obvious way to other situations. He also gives a third characterization, namely, mu is the dimension (over C) of the vectorspace which is the quotient of the ring of formal power series (centered at the critical point) by the ideal generated by f and its first partial derivatives. (I think I have that right.) Other characterizations have been given. > (Favorite factoid in that direction: for any integer k, S^9 intersect > {(z1,...,z5) ; z1^(4k+1) + z2^3 + z3^2 + z4^2 + z5^2 = 0} > is homeomorphic to S^7, and diffeomorphic iff k = 0 mod 28.) Then there's always the neat fact (not, of course, in Milnor's book, since it wasn't knowable at the time) that {(z1,...,z4) ; z1^2 + z2^3 + z3^5 = 0}, a complex hypersurface in C^4 with non-isolated singularities along a complex line L, is homeomorphic to C^3, but the image of L in C^3 by any such homeomorphism is such a mess that its complement isn't simply connected (notice that its real codimension is *four*). LR