I was going through upgrade process of mysql server. It is mentioned that to upgrade from 4.1 to 5.5 we need to upgrade step by step like first to 5.0,5.1 ...5.4 and then 5.5.

I have some questions :


What if we directly upgrade from 4.1 to 5.5. What might get effected? functionality or problem related to upgrading
Chances to loose data ? Even though I'll take backup using 'mysqldump --all-data-base' option.
Lets say, I have two database. I'll take only two database backup. I'll install latest version of mysql 5.7 and load two database back. What are chances to loose data or get corrupted ?


I'm curious to know this. If anybody tried? Because I need to do upgrade in live server.

Is there any step by step document which tells exact steps to upgrade from 4.1 to 5.5 in linux server


Question: Existing proofs of Rokhlin's theorem for PL manifolds
Answer: Another approach to the theorem that could probably be rewritten to work in the PL category is the approach of Kirby and Melvin in Appendix C of the following paper:

MR1117149 (92e:57011) 
Kirby, Robion(1-CA); Melvin, Paul(1-BRYN)
The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C). 
Invent. Math. 105 (1991), no. 3, 473–545. 

See Corollary C6.

The idea of this approach is as follows.  There is a famous $\mathbb{Z}/2$-invariant of homology $3$-spheres called the Rokhlin invariant.  The usual definition of this invariant is as follows.  Letting $M^3$ be a homology $3$-sphere, there exists a compact spin $4$-manifold $W^4$ with $\partial W^4 = M^3$.  Let $\sigma$ be the signature of $W^4$.  Rokhlin's theorem implies that modulo $16$, the value of $\sigma$ is independent of $W^4$.  Since $\sigma$ is divisible by $8$ for number-theoretic reasons (namely, van der Blij's lemma about quadratic forms), the value of $\sigma/8$ is well-defined modulo $2$.  

Using the Kirby calculus, Kirby and Melvin give a $3$-dimensional'' construction of the Rokhlin invariant, avoiding all mention of $4$-manifolds.  They then go backwards and use this to prove Rokhlin's theorem about $4$-manifolds.

Looking at their proof, Kirby and Melvin use smoothness in two ways.  The first is to prove that the Rokhlin invariant is well-defined.  But this is harmless since (by work of Moise) all PL $3$-manifolds can be smoothed in a unique way.  The second use of smoothness is to obtain a handlebody decomposition of the $4$-manifold.  But this should be easier in the PL category!