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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. Phil thesis, defended at the University of Oxford in 2003. 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.

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