Novikov additivity I (Ex)
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Let be a -dimensional manifold with boundary, . Consider the homomorphism and denote the image of . Coefficients are understood to be in .
The middle dimensional intersection form
is degenerate in general. Show that the intersection form on defined by
is a non-degenerate symmetric bilinear form and let us define the signature to be the signature of this form.
Suppose that we have also another -dimensional manifold with boundary . Form the closed manifold . Show that
Observe that the analogous statement is true if we replace manifols with boundary by Poincare pairs.
Hint: section 7 of [Atiyah&Singer1968b]
References
- [Atiyah&Singer1968b] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR0236952 (38 #5245) Zbl 0164.24301