Intersections of loops in two-dimensional manifolds

Given an arbitrary smooth two-dimensional manifold $A$ with nonempty boundary and a point $a\in\partial A$, mappings $\mathbf Z[\pi_1(A,a)]\times\mathbf Z[\pi_1(A,a)]\to\mathbf Z[\pi_1(A,a)]$ and $\pi_1(A,a)\to\mathbf Z[\pi_1(A,a)]$. are constructed. In terms of them the author formulates and proves necessary and sufficient conditions for realizability of an element of the group $\pi_1(A,a)$ by a simple loop, conditions for the realizability of a few elements of $\pi_1(A,a)$ by nonintersecting loops and conditions for realizability of an automorphism of this group by a diffeomorphism $(A,a)\to(A,a)$.
Figures: 5.
Bibliography: 14 titles.