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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Filling the Details #1: “Generalised Cohomology of the projective space via Atiyah-Hirzebruch Spectral Sequence”

So the aim of this article will be to fill out the details of the computation of \mathbb{E}^*(\mathbb{C}P^n) one can find in several books like “Adams’ Stable Homotopy and Generalised Homology” Lemma 2.52.5 page 39 or “Kochman’s Bordism, Stable Homotopy, and Adams Spectral Sequence Proposition 4.3.2 page 132. I feel the amount of details one usually found in these references are quite scarce. It could be a problem, since the topic is rather technical instead. Here is my attempt in filling the details.

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Random Exercise #1 “Isomorphism in cohomology induces Isomorphism in Homology”

I’ve decided to start collecting here some random exercises arose during discussions with my colleagues (usually during the weekly pizza-time) and I found interesting.

Let f\colon X \to Y be a continuous map between topological spaces, and R some coefficients. Let Z be the integers.

 

  1. If H_*(f;R) is an isomorphism, is H^*(f;R) one, too?
  2. If H^*(f,Z) is an isomorphism, is H_*(f; R) one, too (for every R)?

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