Ivancevic_Applied-Diff-Geom

520 Applied Differential Geometry: A Modern IntroductionIn the case of just one operator, this means that σ m (d) is an isomorphismoff the zero section.Recall that the Hodge Theorem states that the cohomology of the complexΓ(Y, E) coincides with the harmonic forms, i.e.,H i (E) = Ker(d i)Im(d i−1 ) ∼ = Ker(∆ i ), where ∆ i = d ∗ i d i + d i−1 d ∗ i−1.Without loss of generality, by passing to the assembled complexE + = E 1 ⊕ E 3 ⊕ · · · , E − = E 2 ⊕ E 4 ⊕ · · · ,we can always think of one elliptic operatorD : Γ(Y, E + ) → Γ(Y, E − ),D = ∑ i(d 2i−1 + d ∗ 2i).The Index Theorem states: Consider an elliptic complex over a compact,orientable, even dimensional manifold Y without boundary. The index ofD, which is given byInd(D) = dim[Ker(D)]− dim[Coker(D)] = ∑ i(−1) i dim[Ker ∆ i ] = −χ(E),χ(E) being the Euler characteristic of the complex, can be expressed interms of characteristic classes as:〈 ∑〉ch(Ind(D) = (−1) n/2 i (−1)i [E i ])td(T Y C ), [Y ] .e(Y )Here, ch is the Chern character, e is the Euler class of the tangent bundleof Y , td(T Y C ) is the Todd class of the complexified tangent bundle. TheAtiyah–Singer Index Theorem, which computes the index of of a family ofelliptic differential operators, is naturally formulated in terms of K−theoryand is an extension of the Riemann–Roch Theorem.4.5.6 The Infinite–Order CaseTopological K−theory turned out to have a very natural link with the theoryof operators in quantum Hilbert space. If H is an infinite–dimensionalcomplex Hilbert space and B(H) the space of bounded operators on H withthe uniform norm, then one defines the subspace F(H) ⊂ B(H) of Fredholmoperators T, by the requirement that both Ker(T ) and Coker(T ) have