I would like to check that the Lefschetz number $\Lambda_f$ and the degree $deg_f$ of a smooth mapping $f:\mathbb{S}^2\rightarrow \mathbb{S}^2$ satisfy $$\Lambda_f-deg_f=1$$ but I am clueless. I know that since $\mathbb{S}^2$ is compact, connected and orientable manifold then the following equations hold $$deg(f)=\sum_{x\in f^{-1}(p)} sign\ df(x)$$ $$\Lambda_f=\sum_{x\text{ fixed point}} sign(df(x)-I)$$ where $p$is a regular value for $f$ and $f$ is a Lefschetz mapping. I guess they may be useful for this purpose, but I cannot relate them to get to the desired equation.
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3This will be far, far easier with the homological definition of Lefschetz number and degree. It is a less-than-one-line argument. – Ted Shifrin Dec 22 '21 at 01:27
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2P.S. You've also omitted important hypotheses here: $p$ is a regular value of $f$, and $f$ is has only Lefschetz fixed points. – Ted Shifrin Dec 22 '21 at 01:35
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Thank you for your answer and remarks! What would be a possible path using these "Linear Algebra" formulas? I would like to know how complicated it is supposed to be if not following your suggestion – beginner_user123 Dec 22 '21 at 01:59
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1I don’t see anything obvious. This is very special to $S^2$. So you can reduce to $\Bbb R^2$. Where did this come from? – Ted Shifrin Dec 22 '21 at 02:36
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From these lecture notes: https://people.math.ethz.ch/~salamon/PREPRINTS/difftop.pdf – beginner_user123 Dec 22 '21 at 02:38
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Exercise 4.4.15 – beginner_user123 Dec 22 '21 at 02:59
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1Note they do have the (co)homological definition here as well. That is their intent. – Ted Shifrin Dec 22 '21 at 04:33