Atiyah-Singer指标定理证明中的部分错误

(b) The Atiyah-Singer index theorem ([AS1], [AS2]) The statement of the so-called “Atiyah-Singer index theorem” would be meaningless without the “Hodge’s theory” for elliptic operators on differentiable vector bundles of a compact complex manifold, see p.142 of [W], and p.423 of [AS1]. In particular, without knowing that whether or not the dimensions of the cohomology groups are finite, then the so-called “Euler characteristic” χ( , ) M E as well as the so-called “Hirzebruch-Riemann-Roch formula” (cf. p.28 of [CG]) are all meaningless. [AS1] (p.431) also used the “cobordism theory”, which has been objected in §1.6 of this memoir. In addition, the so-called “Todd classes” has already been objected in §1.5 of this memoir. In [AB], the authors tried to prove the finiteness of the dimensions of certain homology groups ...  coming from elliptic complexes,by “avoiding Hilbert space entirely”(cf. p.374, [AB]). ...

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（摘自我对黎曼几何批判的§8）