# The Groove of Math

## Filling the Details #3: Homotopy Fibrations and Long Exact sequence in Cohomology

Currently I’m reading the paper of Prof. Peter Teichner On the Signature of four-manifolds with universal covering spin, and I was stuck on the following passage:

The homotopy fibration $\tilde{M} \to M \to K(\pi, 1)$ induces an exact sequence in cohomology

$0 \to H^2(\pi ; Z/2) \to H^2(M;Z/2) \to H^2( \tilde{M} ; Z/2)$

I managed to find a solution to it now, and I thought it could be a good idea to write it here:

## 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

## 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 $\tau_A|M$.

## 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.

## 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 $R$)?