The Groove of Math

or in other words, the Devil is in the Details


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Filling the Details #2: Representing 2-homology classes of a 4-manifold.

For the second article under the FtD tag, I’ve chosen a nice proposition which bothered me some times ago. It’s intuitively easy to believe but I found a scarce amount of details spelled out in the proofs one can find on the classical references. So here we have the following result:

Proposition: Every 2-homology class of a smooth 4-manifold is generated by a surface

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Random Exercise #2: Problem 11-C in Milnor’s Characteristic Classes

Todays’ random exercise will be the pretty famous exercise 11-C  one can find in Milnor’s Book. Let me recall the statement:

Let M = M^n and A = A^p be compact oriented manifolds with smooth embedding i : M \rightarrow A. Let k = p-n. Show that the Poincare duality isomorphism  \frown \mu_A : H^k(A) \rightarrow H_n(A) maps the cohomology class u^{'}|A dual to M to the homology class (-1)^{nk} i_{*} (\mu_M). Assume moreover that the normal bundle \nu^k is oriented so that \tau_M \oplus \nu^k is orientation preserving isomorphic to \tau_A|M.

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