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Extra info for A K-Theory Proof of the Cobordism Invariance of the Index

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Using Atiyah and Singer’s fundamental result that, on closed manifolds, the cobordism invariance of the index of twisted signature operators yields the index formula, one has then that these two results are in fact equivalent on closed manifolds. It is not clear if the same is true for general σ -compact manifolds and pseudodifferential operators that are multiplication at infinity. 9 from the cobordism invariance of the index. 2. Let M, W be manifolds, i : M → W be an embedding and consider the push-forward map i!

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! (σ )). Acknowledgements This work is part of a D.

I would like to thank my supervisors Prof Ulrike Tillmann and Prof Victor Nistor, from the Pennsylvania State University, for their encouragement and support, and my thesis examiners for useful comments. ¨ I would also like to thank Klaas Landsman and Michael Muger for very helpful suggestions. References 1. 2. : K-theory. Benjamin, New York, 1967. Atiyah, M. : The index of elliptic operators on compact manifolds, Bull. Am. Math. Soc. 69 (1963), 422–433. 3. Atiyah, M. : The index of elliptic operators I, Ann.

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