I am trying to understand the connection between The W-hierarchy as presented in chapter 13 of this book by Cygan et al. and the notion of the NP problems.
Is the existence of an FPT algorithm for a problem in W[1] suggests that P=NP? Why?
For example, assuming I have an FPT algorithm for the k-Clique problem. Can I prove the P=NP? The algorithm run time will still be exponential. Only now, it will depend on $k$.
Because $k$-CLIQUE is W[1]-hard, your result would imply that FPT = W[1], which by the result of Downey and Fellows [1] implies that the Exponential Time Hypothesis (ETH) is false, i.e., that 3-SAT can be solved in $2^{o(n)}$ time. But ETH is stronger than P not equal to NP, so its falsification doesn't settle P = NP (but of course, if true, ETH would imply that P is not equal to NP).
As far as I know, FPT = W[1] is not known to imply anything else regarding the W-hierarchy.
[1] Downey, Rodney G., and Michael Fellows. Parameterized complexity. Springer Science & Business Media, 2012.
If $\mathrm{FPT}=W[1]$, then we will not directly get an answer to $\mathrm{P}$ vs. $\mathrm{NP}$ (or at least, not that we know). The assumption that $\mathrm{FPT} \neq W[1]$ is a weaker claim than $\mathrm{P}\neq \mathrm{NP}$:
If $\mathrm{P} = \mathrm{NP}$, then there is polynomial time algorithm for the $W[1]$-complete problem $k$-Clique, so $\mathrm{FPT}= W[1]$. Taking the contrapositive, we get that $\mathrm{FPT}\neq W[1]$ implies $\mathrm{P}\neq \mathrm{NP}$.
Taking a step back, the entire concept of fixed parameter tractability implicitly assumes that there are problems for which we cannot obtain a polynomial time algorithm. By applying this concept to NP-complete problems, we make the assumption that we are unable to prove $\mathrm{P} = \mathrm{NP}$. While this argument does not rule out a proof of $\mathrm{FPT}=W[1]$ implies $\mathrm{P} = \mathrm{NP}$, it does not seem natural that a collapse of complexity classes designed to give more fine-grained results when $\mathrm{P}\neq \mathrm{NP}$ would imply $\mathrm{P} = \mathrm{NP}$.
