Novikov additivity II (Ex)

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This is an exercise in Novikov additivity for normal complexes.

Let (Y,X) be a 2n-dimensional Poincare pair with the fundametal class [Y] which maps to the fundamental class [X] under the boundary homomorphism, n = 2k.

Suppose that we have a 2n-dimensional normal pair (Y',X) whose fundamental class [Y'] maps to -[X] under the boundary map. Form the normal space Z = Y' \cup_{X} Y. Show that

\displaystyle  \partial \textup{sign}^{\mathbf{NL}^{\bullet}} (Z) = \partial \textup{sign}^{(\mathbf{NL}^{\bullet},\mathbf{L}^{\bullet})} (Y',X)

Hint: check the Novikov additivity exercise and the definition of the quadratic boundary of a (normal,Poincare) pair.

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