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My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave.

This is the definition of local index, Theorem 11.9 and the beginning of the proof of Theorem 11.9, which refers to Lemma 11.8.

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In the beginning of the proof, I think it is claimed that $f|_{V_{q_i}}$ is positively oriented if and only if $D_qf$ is orientation-preserving.

  1. Do I understand right?

  2. If not, then what is meant? If so, then is the following proof correct?

    • 2.1 In the first place, $(V_{q_i},f_{V_{q_i}})$ is a coordinate chart about $q_i$ in $N$, by Proposition 6.11 in An Introduction to Manifolds by Loring W. Tu, and therefore it makes sense to talk about whether or not $f|_{V_{q_i}}$ is positively oriented since I guess "positively oriented" is for coordinate maps and not arbitrary diffeomorphisms of manifolds...unless "positively oriented" actually means orientation-preserving, so the use of the adjective "positively oriented" for charts actually comes from the use of "positively oriented" for arbitrary diffeomorphisms of manifolds.

      • Update: It's not a chart, as pointed out in answer. I forgot that I had actually realized that earlier.

    • 2.2 $f|_{V_{q_i}}$ is positively oriented if and only if $D_q(f|_{V_{q_i}})$ is orientation-preserving.

      • I'm not too sure about what the definition of a positively oriented chart is (see here and here), but what I'm hoping is that whatever the definition is, (2.2) is an equivalent definition.

    • 2.3 $D_q(f|_{V_{q_i}})$ and $D_q(f)$ are identical by chain rule and by the fact that differential of inclusion from an open subset is still inclusion

    • 2.3 Therefore, by (2.2) and (2.3), $f|_{V_{q_i}}$ is positively oriented if and only if $D_qf$ is orientation-preserving.

    • 2.4 Remark: I think we don't assume connectedness here.

  3. Do we have that $f|_{V_{q_i}}$ is negatively oriented if and only if $D_qf$ is orientation-reversing even if $U$ isn't connected?

Some context:

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If you have a diffeomorphism $\phi : A \to B$ between oriented manifolds, then all $D_q\phi : T_qA \to T_{\phi(q)}B$ are isomorphisms of oriented vector spaces. This gives you a function $\sigma_\phi : A \to \{+1, -1 \}$ defined by $\sigma_\phi(q) = +1$ iff $D_q\phi$ is orientation preserving. This function is locally constant, hence constant on each connected component $C$ of $A$. This means that $\phi \mid_C$ is either orientation preserving or orientation reversing. But note that on each connected component $\sigma_\phi$ may take an individual value. In other words, for a non-connected $A$ you may have the situation that $\phi$ is neiter orientation preserving nor orientation reversing.

This is the reason why $U$ is assumed to be connected. In that case you can moreover say that $f \mid_{V_i}$ is orientation preserving iff $D_qf$ is orientation preserving.

For your point 3. the answer is yes only for connected $U$.

Paul Frost
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  • Is (2.4) wrong? My understanding is as follows 2.4.a. for connected $U$, $f|{V{q_i}}$ must be either positively or negatively oriented and $D_q(f|{V{q_i}})$ must be either orientation-preserving or orientation-reversing, and 2.4.b. that $f|{V{q_i}}$ is positively (negatively) oriented if and only if $D_q(f|{V{q_i}})$ is orientation-preserving (orientation-reversing). –  May 08 '19 at 10:11
  • 2.4.c. For disconnected $U$, $f|{V{q_i}}$ may be neither positively or negatively oriented. 2.4.d. $D_q(f|{V{q_i}})$ may be neither orientation-preserving or orientation-reversing. 2.4.e. However, whenever, which may be never, we have that $f|{V{q_i}}$ is positively (negatively) oriented, we have that $D_q(f|{V{q_i}})$ must be orientation-preserving (orientation-reversing), and the converse is true. –  May 08 '19 at 10:13
  • For (3) to clarify, my understanding (this is kind of similar to (2.4.a-e) in comments above) is that for disconnected $U$, we have that "$f_{V_{q_i}}$ is not positively oriented" is not equivalent to saying that "$f_{V_{q_i}}$ is negatively oriented", so what is wrong with saying that whenever, which may be never, $f_{V_{q_i}}$ is negatively oriented, we have that $D_q(f|{V{q_i}})$ is orientation-reversing, or conversely? –  May 08 '19 at 10:17
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    Although I do not know the book by Madsen and Tornehave, it is obvious that $f \mid_{V_k} : V_k \to U$ is not a chart on $N$ since $U \subset M$, but a diffeomorphism between manifolds (which are open subsets of $N, M$). Hence the wording "$f \mid_{V_k}$ is positively (negatively) oriented" must be a synonym for "$f \mid_{V_k}$ is orientation preserving (reversing)". This is a reasonable generalization of being "positively (negatively) oriented" from charts on oriented manifolds to diffeomorphisms between oriented manifolds. – Paul Frost May 08 '19 at 14:25
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    A single $D_q(f \mid_{V_k})$ is always orientation-preserving or orientation-reversing, but only for connected $U$ one can conclude that the whole diffeomorphism $f \mid_{V_k}$ is orientation-preserving or orientation-reversing. – Paul Frost May 08 '19 at 14:26
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    But be aware that notation is frequently not standardized in the literature. So you have to be careful when reading different books at the same time. – Paul Frost May 08 '19 at 14:29