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Hi, I want to find an example of a Morse-Bott function such that for at least one of the critical submanifolds, the orientation sheaf O is nontrivial.

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Any non-orientable vector bundle over a manifold will do the trick. Put a Riemann metric on the vector bundle and make your Morse function the distance squared from the origin. Hutching's example is the lowest-dimensional possibility. –  Ryan Budney Jan 17 '12 at 2:20
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2 Answers 2

You can find such a function on the Klein bottle. If you think of the Klein bottle as $S^1\times [0,1]$ with the two boundary components glued together via a reflection $\phi$, and if $f:S^1\to {\mathbb R}$ is a Morse function with two critical points which is invariant under $\phi$, then $f$ extends to a Morse-Bott function on the Klein bottle such that the circle of maxima has a nontrivial orientation sheaf.

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How can you extend $f$ to a Morse-Bott function on the Klein bottle such that the circle of maxima has a nontrivial orientation sheaf? –  zatilokum Jul 27 '12 at 12:18
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You can get a large class of examples if you look at the real projective space $\mathbb{RP}^n$. Each point $L\in \mathbb{RP}^n$ is a one dimensional subspace in $\mathbb{R}^{n+1}$ and we denote by $P_L$ the orthogonal projection onto $L$. Fix a unit vector $v\in \mathbb{R}^{n+1}$ and denote by $U$ the orthogonal complement in $\mathbb{R}^{n+1}$ of the line $L_v$ spanned by $v$. Consider the function

$$f_v: \mathbb{RP}^n\to\mathbb{R}, \;\; f_v(L)=(P_Lv,v). $$

Note that

$$ 0 \leq f_v(L)\leq 1,\;\;\forall L.$$

This is a Morse-Bott function with precisely two critical submanifolds: the locus of minima where $f=0$ and consisting of the real projective space $\mathbb{P}(U)\subset \mathbb{RP}^n $ of lines in $U$, and a unique maximum point, $L_v\in\mathbb{RP}^n$ where $f_v(L_v)=1$.

The normal bundle of $\mathbb{P}(U)\subset \mathbb{RP}^n $ is the tautological real line bundle over $\mathbb{P}(U)$ which is nonorientable.

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