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Nicolaescu, L. : On the cobordism invariance of the index of Dirac operators, Proc. Am. Math. Soc. 125(9) (1997), 2797–2801. Palais, R. ): Seminar on the Atiyah-Singer Index Theorem. Annals of Math. Studies, 57. Princeton University Press, Princeton, New Jersey, 1965. Shubin, M. : Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, Heidelberg, 2001. : Pseudodifferential Operators, Princeton Mathematical Series 34, Princeton University Press, Princeton, New Jersey, 1981. : Introduction to Pseudodifferential and Fourier Integral Operators, Plenum Press, New York, 1980.

2. Let M, W be manifolds, i : M → W be an embedding and consider the push-forward map i! : K 0 (T M) → K 0 (T W ). If there is a manifold Y with ∂Y = M W and an embedding j : M × I → Y such that j|M×{0} = id and j|M×{1} = i, then ind ◦i! = ind. Proof. 2 to the embedding i : M M → M W given by i := id i. In this case, one easily checks that i! = id ⊕ i! and, since (M M, −σ ⊕ σ ) ∼ 0, through a cobordism (M × I, ω), we have also that (M W, −σ ⊕ i! (σ )) ∼ 0, that is, (M, σ ) ∼ (W, i! (σ )). From the cobordism invariance of the index, we conclude that ind(σ ) = ind(i!

Nicolaescu, L. : On the cobordism invariance of the index of Dirac operators, Proc. Am. Math. Soc. 125(9) (1997), 2797–2801. Palais, R. ): Seminar on the Atiyah-Singer Index Theorem. Annals of Math. Studies, 57. Princeton University Press, Princeton, New Jersey, 1965. Shubin, M. : Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, Heidelberg, 2001. : Pseudodifferential Operators, Princeton Mathematical Series 34, Princeton University Press, Princeton, New Jersey, 1981. : Introduction to Pseudodifferential and Fourier Integral Operators, Plenum Press, New York, 1980.