In this work, we present the Atiyah-Singer index theorem together with the necessary mathematical background. We begin by developing $K$-theory, including reduced, relative, and negative $K$-groups, compactly supported $K$-theory, external products, complexes of vector bundles, Bott periodicity, and the Thom isomorphism. These tools are used to construct the first of two key ingredients of the Atiyah-Singer index theorem, the topological index $t$-ind.
The analytic side begins with pseudodifferential operators on ${\mathbb R}^n$, which are then extended to operators between sections of vector bundles over manifolds. After introducing Sobolev spaces and ellipticity, we prove that elliptic pseudodifferential operators on closed manifolds are Fredholm. The principal symbol determines a correspondence between such operators and classes in $K_c(T^\vee M)$, which allows us to define the second key ingredient of the Atiyah-Singer index theorem, the analytic index $a$-ind.
We then formulate the theorem as the equality $t$-ind $=a$-ind. We explain the formal structure of the proof and refer to the relevant literature for the technical details. Finally, using the Chern character, we translate the $K$-theoretic statement into a cohomological formula and derive the well-known formulas $\chi(M)=\langle e(TM),[M]\rangle$ and $
{\rm sign}(M)=\langle L(TM),[M]\rangle$ as corollaries.
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