# The Groove of Math

## Random Exercise #5: The homotopy fibre of CP^1 ↪ CP^∞ and a Serre Spectral Sequence

[It is really annoying that the title cannot contain any tex code:( ]

Hi all, sorry for not having written much in the blog lately, i was kind of busy. First of all, I’m happy to announce that next I’ll be a graduate student at UT in Austin. I’m so happy and I’m looking forward to start working there. Ok, back to business, on today’s random exercise I will deal with a problem I encountered today. Since I found it funny and interesting I will write it down here. We are asked to study the homotopy fibre of the map

$\mathbb{C}P^1 \hookrightarrow \mathbb{C}P^{\infty}$.

## Random Exercise #4: The nonexistent 5-Manifold in Bredon’s book

Dear all, sorry for not having posted anything so far but my master thesis (stable classification of certain families of four manifolds) and PhD interviews have been keeping me very busy 🙂

This is a quick exercise which I was asked to solve by a friend of mine and since I found it funny I decided to spell some more details about it in order to show the importance of the Bockstein exact sequence. This could be a typical exam exercise in “Topology 2” here in Bonn, so dear fellows students, read and comment if anything is unclear!

The following is exercise 7, page 366 from Bredon’s Topology and Geometry:

Show that there can be no $5$-manifold $M$ with

## Filling the Details #4: Maps of Eilenberg MacLane Spectra induces stable cohomology operations

First of all, sorry for not having written lately. I’ve just done the GRE and GRE math examinations, together with the TOEFL. I really hope the result will be good, since I’m going to apply in the USA!

Ok, back to Math 🙂  As the title suggests, I want to write something about this known result, since I needed it for understanding why the first non-vanishing differential in the Atiyah-Hirzebruch Spectral Sequence is a stable operation. What I’m going to show is not the nicest way to prove it, but still I think it’s worth to take a look at a more “concrete” proof

## 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:

## 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 $\tau_A|M$.

## 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 $R$)?