Some possible mistakes in Bott and Tu [closed]

I've checked five of your claimed errors, none of which you're right about. There are other cases in which you've found obvious typos, which are admittedly annoying but which shouldn't cause any real trouble.

It may be you've found some significant errors in the other points which are more complicated for me to check, but based on the following evidence it looks like you need to take the authors more seriously and check your objections more critically. As a bit of unasked for advice, I should point out that Bott and Tu is one of the most important and valuable books in the subject, and it seems likely that you're losing something in making your reading of it a contest with the authors. On the other hand, it does seem to be leading to you reading more closely than many would, so perhaps there's no harm done.

1) You need to think more carefully about what geodesic convexity is for the circle.

2) The kernel of a map of vector bundles collapsing one fiber is not a vector bundle. Given that your sequence is a sequence of vector bundles, this is fine.

5) No, they'd be claiming the inclusion $\mathbb{R}^n\setminus\{0\}\to \mathbb{R}^n$ is homotopic to a constant, which is obviously true since every map to $\mathbb{R}^n$ is.

9) No, there's a difference between a constant sheaf and a constant presheaf.

13) This is something you can easily check for yourself. For instance when $k=2$ it amounts to claiming that I can extend every map from a circle to a map from the disk when my space is simply connected. Indeed, a nullhomotopy of a map from the circle is exactly the same thing as a map from the disk.