A Family of Nontrivial Homology Classes (part 2)
We continue investigating the differential forms we defined last time. Recall that we started with the position vector field and use the interior product to produce the -form on the punctured -dimensional space . We restrict this form to the -dimensional sphere and then pull back along the retraction mapping to get the form .
I’ve asserted that , and now we will prove it; let be tangent vectors at and calculate
as asserted. Along the way we’ve used two things that might not be immediately apparent. First: the derivative works by transferring a vector from to and scaling down by a factor of , which is a consequence of the linear action of and the usual canonical identifications. Second: the volume form on can be transferred to essentially the same form on itself by using the canonical identification .
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