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I disagree that index theorem is "one of the most important results in twentieth century's mathematics". This kind of superlatives usually does not mean anything, unless one can draw a fine line between "important", "more important", "most important", "of greatest importance", etc. I think it is better to be humble about it, even though my own research field ...


From the Wikipedia page on the index theorem: One can also define the topological index using only K theory (and this alternative definition is compatible in a certain sense with the Chern-character construction above). If $X$ is a compact submanifold of a manifold $Y$ then there is a pushforward (or "shriek") map from $K(TX)$ to $K(TY)$. The topological ...

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