Timeline for Extending the natural thom form of a vector bundle from the boundary of a manifold

5 events

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Apr 15 at 11:16	history	edited	Kai Hugtenburg	CC BY-SA 4.0	deleted 103 characters in body; edited title
Apr 15 at 11:06	comment	added	Kai Hugtenburg		Thanks @TomGoodwillie, for some reason I had convinced myself that partition of unity arguments wouldn't work, as we are trying to obtain a closed form. However, what you write works! The second part of the question is indeed different from the first question. For my purposes your answer is sufficient, but the second part is hopefully of independent interest. I will edit the question to make this clear.
Apr 15 at 10:42	comment	added	Tom Goodwillie		For the question as stated in the first sentence, the answer is "yes". (But maybe you really want to use a metric and a connection.) If $\sigma$ represents the Thom class of $E$ and $\tau$ represents the Thom class of $i^\ast E$, then the difference between $\tau$ and the restriction of $\sigma$ is $d\rho$ for some form $\rho$ (with the appropriate kind of support), so extend $\rho $ to $E$ and add the differential of that to $\sigma$ to get a Thom form that restricts to $\tau$.
S Apr 15 at 10:23	review	First questions
Apr 15 at 11:27
S Apr 15 at 10:23	history	asked	Kai Hugtenburg	CC BY-SA 4.0