I need to determine whether the following maps are ring homomorphisms (there were actually 5 to begin with, but I was able to figure out 2 of them on my own). In addition, if yes, I need to determine whether they are also monomorphisms, epimorphisms, or isomorphisms.
The map $\mathbb{Z}(G...
Can anyone help me with a general polar equation? I'm given a circle with polar coordinates, say r = xsin(theta) + ycos(theta). There is a region shaded on the circle from [0,2]. If we revolve the region about the y-axis, what would be the volume of the solid generated? With the disk/washer method.
It's not so much about the answer, but rather the process that I
@PVAL The version I mean is that a foliated neighborhood is determined by the germinal holonomy of the leaf (and indeed can just be written in terms of the holonomy). The compactness assumption implies the image of the holonomy in germs of homeomorphisms of (R,0) can only be the identity and negation.
@PVAL Well Mike set that as an exercise so he probably already knows.
On second thought I don't understand PVAL's objection anymore. A small, open, tubular neighborhood - once made transverse to the leaves - is a neighborhood of the leaf which is a union of the nearby leaves, not?
I don't remember why we cared about the boundary of the tubular neighborhood in the first place
@MikeMiller Do you need to know that once you understand there's a normal neighborhood foliated by nearby leaves? I mean then it's isomorphic to either the trivial line bundle on L foliated obviously or the "twisted" line bundle on L foliated by small S^0 bundles.
@MikeM Anyway, the reason I asked this initially is because that proves your exercise. A small transversal is then hit by leaves in antipodes, more or less. That's the Z/2 action by reflection and gives it to you as a bundle over the orbifold over that.
@Akiva .pdf's are not opening in this end (slow internet) but the exposition is by Matthew Bond, "Convolutions and The Weierstrass Approximation Theorem"