# 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

## Random Exercise #3: K(G,1) is a model for the classifying space BG via covering theory (NOT l.e.s. of a fibration)

I decided to write down this nice exercise since it is the kind of exercise which is readily solved with the right tools (i.e. l.e.s. of a fibration), but can become non-trivial if one doesn’t know them.

In particular, I want to prove that the Eilenberg MacLance space $K(G,1)$, for a discrete group $G$ is a model for the classifying space $BG$.

## 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 $R$)?