Before we go too far into the world of spectra and generalized cohomology, I want to catalog some nice theorems that I have used but haven’t studied much in the past.  Let’s start with the Freudenthal suspension theorem, which nicely falls out of the Serre spectral sequence.  It will follow from the following proposition.

Proposition: If $X$ is an $n$-connected CW-complex, then the canonical map $j: X \rightarrow \Omega \Sigma X$ is a $(2n+1)$-equivalence.

Proof: Consider the path-loop fibration $\Omega \Sigma X \rightarrow P \Sigma X \rightarrow \Sigma X$.   By the Hurewicz theorem, $H_k(X)=0$ for $k \leq n$.  Since suspension induces an isomorphism on homology, we have that $H_k(\Sigma X)=0$ for $k \leq n+1$.  By the Hurewicz theorem again, we have $\pi_k(\Sigma X)=0$ for $k \leq n+1$.  By the adjointness of loops and suspension, we have $\pi_k(\Omega \Sigma X)=\pi_{k+1}(\Sigma X)$, which is zero for $k \leq n$.  So in the above fibration, the fiber is $(n+1)$-connected and the base is $n$-connected.  Looking at the Serre spectral sequence, we see that for $k \leq 2n+1$, the first nonzero differential from $H_k(\Sigma X)$ and the first nonzero differential into $H_{k-1}(\Omega \Sigma X$ must be the transgression, $\tau$.  Since $P\Sigma X$ is contractible, $\tau$ is an isomorphism for $k \leq 2n+1$.  Now we have the following diagram

(diagram coming soon–I need to figure out how to draw commutative diagrams on this blog and in TeX in general)

where $J: CX \rightarrow P\Sigma X$ is given by $J(u, x)(t)=(tu, x)$ where $CX$ is the cone on $X$ and $q$ is the map that collapses $X$ to a point.  Notice that the composition $X \rightarrow CX \rightarrow \Sigma X$ is exactly the suspension.  Going to the long exact sequences in homology, we have

(diagram coming soon)

where the composition of $latex$\delta\$ and $q_*$ is exactly the suspension isomorphism on homology.

TO BE COMPLETED…

Freudenthal Suspension Theorem: Let $X$ be an $n$-connected CW-complex.  Then $S: \pi_k(X) \rightarrow \pi_{k+1}(\Sigma X)$ is an isomorphism for $k \leq 2n$.

Proof: We have $[S^k, X] \rightarrow [S^k, \Omega \Sigma X] \rightarrow [\Sigma S^{k+1}, \Sigma X]$ for $k \leq 2n$$\Box$

More theorems will be added soon: Blakers-Massey, how to get long exact sequences from fibrations, etc.

Next week I plan on writing up some notes on complex oriented cohomology theories and the connection to formal group laws.  I will also continue studying spectra on the way to the Adams spectral sequence.  So far, I’ve been getting most of my notes from the book Bordism, Stable Homotopy, and Adams Spectral Sequences by S.O. Kochman.