The Atiyah-Singer index theorem states that the analytic and topological "indices" are equal for any elliptic differential operator on an compact smooth manifold .
For their discovery and proof is this theorem, Atiyah and Singer shared the 2004
Abel prize .
See also Compact Manifold ,
Smooth
Manifold
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References Atiyah, M. F. and Singer, I. M. "The Index of Elliptic Operators on Compact Manifolds." Bull. Amer. Math. Soc. 69 ,
322-433, 1963. Atiyah, M. F. and Singer, I. M. "The Index
of Elliptic Operators I, II, III." Ann. Math. 87 , 484-604, 1968. Petkovšek,
M.; Wilf, H. S.; and Zeilberger, D. A=B. https://www2.math.upenn.edu/~wilf/AeqB.html . Rognes,
J. "On the Atiyah-Singer Index Theorem." https://abelprize.no/sites/default/files/migrated-media/Atiyah-%26%23151%3BSinger%20index%20theorem.pdf . Referenced
on Wolfram|Alpha Atiyah-Singer Index Theorem
Cite this as:
Weisstein, Eric W. "Atiyah-Singer Index Theorem."
From MathWorld https://mathworld.wolfram.com/Atiyah-SingerIndexTheorem.html
Subject classifications
-dimensional compact smooth 
boundaryless manifold.
