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Could you help me to see why signature is a HOMOTOPY invariant? Definition is below (from Stasheff) The \emph{signature (index)} $\sigma(M)$ of a compact and oriented $n$ manifold $M$ is defined as follows. If $n=4k$ for some $k$, we choose a basis $\{a_1,...,a_r\}$ for $H^{2k}(M^{4k}, \mathbb{Q})$ so that the \emph{symmetric} matrix $[]$ is diagonal. Then $\sigma (M^{4k})$ is the number of positive diagonal entries minus the number of negative ones. Otherwise (if $n$ is not a multiple of 4) $\sigma(M)$ is defined to be zero \cite{char}.

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    Cohomolgy groups are diffeomorphism invariants, but ?2012-06-16
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    Cohomology groups are homotopy invariants.2012-06-16
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    Cohomology groups are not diffeomorphism invariants! For example, Milnor's exotic spheres show this. Although I am sure there are other easier examples. For example, take any smooth thickening of a manifold...2012-06-18
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    Note that the signature is an *oriented* homotopy invariant, i.e., it is invariant only under orientation preserving homotopies.2013-03-03
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    @SeanTilson: Cohomology groups are invariant under homotopy equivalence, and thus under homeomorphism, and thus under diffeomorphism. Milnor's exotic spheres are all homeomorphic--- and thus, in particular, cohomology spheres--- but not diffeomorphic.2015-08-17
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    Yes, but they do not distinguish between different diffeomorphism classes. I guess I was less than clear by what I meant.2015-08-17

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