Algebraic structures on generalized strings

A garland based on a manifold $P$ is a finite set of manifolds homeomorphic to $P$ with some of them glued together at marked points. Fix a manifold $M$ and consider a space $\text{NN}$ of all smooth mappings of garlands based on $P$ into $M$. We construct operations $\bullet$ and $[-,-]$ on the bordism groups ${\text{bor}}_{\ast }(\text{NN})$ that give ${\text{bor}}_{\ast }(\text{NN})$ the natural graded commutative assosiative and graded Lie algebra structures. We also construct two auto-homomorphisms $\text{pr}$ and $\text{li}$ of ${\text{bor}}_{\ast }(\text{NN})$ such that $\pr(\li \alpha_1\bullet \li \alpha_2)= [\alpha_1, \alpha_2]$forall$\alpha_1, \alpha_2 \in \bor_*(\NN)$\mathrm{.}If$P$isaboundary,then$\pr \circ \li=0$ and thus ${\mathrm{\Delta }}^2=0$ for $\mathrm{\Delta }=\text{li}\circ \text{pr}$. We show that under certain conditions the operations $\mathrm{\Delta }$ and $\bullet$ give rise to Batalin-Vilkoviski and Gerstenhaber algebra structures on ${\text{bor}}_{\ast }(\text{NN})$. In a particular case when $P=S^1$, the algebra ${\text{bor}}_{\ast }(\text{NN})$ is related to the string-homology algebra constructed by Chas and Sullivan.