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The fixed point theorem of Atiyah-Bott via parabolic operators. (English) Zbl 0181.36802

Let \(\rightarrow \Gamma(E_i) \overset{d_i}{\longrightarrow} \Gamma(E_{i+1})\rightarrow\) be a general elliptic complex, each of the \(\Gamma(E_i)\) being equipped with a prehilbertian structure so that we have therewith the adjoint elliptic complex. Then, as in the theory of harmonic forms, defining \(H_i\) to be the harmonic projection operator in \(\Gamma(E_i)\), it can be shown that there exists a sequence of degree decreasing operators \(Q_i\colon \Gamma(E_{i+1})\to\Gamma(E_i)\) which permits us to decompose the identity operator \(I_i\) in \(\Gamma(E_i)\) as \(I_i = H_i + d_{i-1}\cdot Q_{i-1} + Q_i\cdot d_i\).
This is a basic lemma for the proof of the fixed point theorem of Atiyah-Bott. In fact, the proof of the theorem can be thereby reduced to a simple local problem of partial differential equations. In this paper, those above are proved in a quite straightforward elementary way.
Reviewer: Takeshi Kotake

MSC:

58J10 Differential complexes
58J20 Index theory and related fixed-point theorems on manifolds
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References:

[1] Atiyah, Ann. of Math. 86 pp 374– (1967)
[2] Eidelman, Mat. Sb. N.S. 38 pp 51– (1956)
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