On homotopy braids

The homotopy braid group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mover accent="true"> <m:mi>B</m:mi> <m:mo>^</m:mo> </m:mover> <m:mi>n</m:mi> </m:msub> </m:math> {{B}ₙ} is the subject of the paper. First, the linearity of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mover accent="true"> <m:mi>B</m:mi> <m:mo>^</m:mo> </m:mover> <m:mi>n</m:mi> </m:msub> </m:math> {{B}ₙ} over the integers is proved. Then we prove that the group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mover accent="true"> <m:mi>B</m:mi> <m:mo>^</m:mo> </m:mover> <m:mn>3</m:mn> </m:msub> </m:math> {{B}₃} is torsion free.