Date: Fri, 3 Jul 1998 11:35:48 -0400 (EDT) From: James Stasheff Subject: query Does anyone know/recall: I want to know is what motivated Hopf for the 1931 paper. Was he specifically looking for nontrivial (non-null-homotopic) maps between spheres? Or was he doing something else and stumbled upon the map S^3 -> S^2 which he then realized had interesting topological properties? cf. Dirac who ``stumbled upon the map S^3 -> S^2'' why investigating the mag monopole - though noone noticed what Dirac had found in topolgoical tems for some decades. ************************************************************ Until August 10, 1998, I am on leave from UNC and am at the University of Pennsylvania Jim Stasheff jds@math.upenn.edu 146 Woodland Dr Lansdale PA 19446 (215)822-6707 Date: Fri, 3 Jul 1998 13:28:25 -0400 (EDT) From: James Stasheff Subject: query from the East A friend asks: Take a smooth fiber bundle with a closed connected manifold as total space and the real Grassmann manifold $G_k(R^n)$ of all $k$-dimensional vector subspaces in $R^n$ as fiber. Then I can relatively easily prove that the fiber is totally nonhomologous to zero rel $Z_2$ (that is, the fiber inclusion into the total space induces an epimorphism in $Z_2$-cohomology) if $n$ is odd and $k$ is arbitrary. But perhaps this also follows from some more general result about fibrations (not necessarily fiber bundles) with a homogeneous space as fiber, and can perhaps also be proved when $n$ is even and $k$ is odd (together with the case of $n$ odd and $k$ arbitrary, this would then cover all those Grassmannians with nonvanishing Euler-Poincare characteristic)? I know of Thm C in Shiga and Tezuka, Rational fibrations etc., Ann. Ins. Fourier 38 (1987), 81-106 (you reviewed it for Math Reviews). But their condition that the order of the corresponding Weyl group should not be divisible by $2$ (in this case) is not satisfied. On the other hand, they also have some remark in the sense (I don't have the paper at hand just now) that perhaps their divisibility condition is not the best possible. Did then someone later prove their Thm C under a weaker (or none?) divisibility condition? ************************************************************ Until August 10, 1998, I am on leave from UNC and am at the University of Pennsylvania Jim Stasheff jds@math.upenn.edu 146 Woodland Dr Lansdale PA 19446 (215)822-6707 Date: Fri, 3 Jul 1998 15:23:33 -0400 (EDT) From: James Stasheff Subject: draft I have posted to my home page math.unc.edu/Faculty/jds Survey of Cohomological Physics It is a ?semi?final draft. Any and all comments appreciated. ************************************************************ Until August 10, 1998, I am on leave from UNC and am at the University of Pennsylvania Jim Stasheff jds@math.upenn.edu 146 Woodland Dr Lansdale PA 19446 (215)822-6707