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The present paper is the notes of a mini-course addressed mainly to non-experts. Its purpose is to provide a first approach to the theory of mapping class groups of non-orientable surfaces.

The purpose of this paper is to put together a large amount of results on the $K(\pi ,1)$ conjecture for Artin groups, and to make them accessible to non-experts. Firstly, this is a survey, containing basic definitions, the main results, examples and an historical overview of the subject. But, it is also a reference text on the topic that contains proofs of a large part of the results on this question. Some proofs as well as few results are new. Furthermore, the text, being addressed to non-experts, is as...

Let $M$ be a surface, let $N$ be a subsurface, and let $n\le m$ be two positive integers. The inclusion of $N$ in $M$ gives rise to a homomorphism from the braid group $B_{n}N$ with $n$ strings on $N$ to the braid group $B_{m}M$ with $m$ strings on $M$. We first determine necessary and sufficient conditions that this homomorphism is injective, and we characterize the commensurator, the normalizer and the centralizer of ${\pi }_{1}N$ in ${\pi }_{1}M$. Then we calculate the commensurator, the normalizer and the centralizer of $B_{n}N$ in $B_{m}M$ for large surface braid...

We define the singular Hecke algebra $\mathscr{H}\left(S{B}_n\right)$ as the quotient of the singular braid monoid algebra $ℂ\left(q\right)\left[S{B}_n\right]$ by the Hecke relations ${\sigma }_k^2=(q-1){\sigma }_k+q$, $1\le k\le n-1$. We define the notion of Markov trace in this context, fixing the number $d$ of singular points, and we prove that a Markov trace determines an invariant on the links with $d$ singular points which satisfies some skein relation. Let ${\mathrm{TR}}_d$ denote the set of Markov traces with $d$ singular points. This is a $ℂ(q,z)$-vector space. Our main result is that ${\mathrm{TR}}_d$ is of dimension $d+1$. This result is completed...