The following preprint is available: M. Grandis, Collared cospans, cohomotopy and TQFT (Cospans in Algebraic Topology, II) Dip. Mat. Univ. Genova, Preprint 555 (2007). http://www.dima.unige.it/~grandis/wCub2.pdf http://www.dima.unige.it/~grandis/wCub2.ps Abstract. Topological cospans and their concatenation, by pushout, appear in the theories of tangles, ribbons, cobordism, etc. Various algebraic invariants have been introduced for their study, which it would be interesting to link with the standard tools of Algebraic Topology, (co)homotopy and (co)homology functors. Here we introduce collared cospans between topological spaces, as a generalisation of the cospans which appear in the previous theories. Their interest lies in the fact that their concatenation is realised with homotopy pushouts. Therefore, cohomotopy functors induce 'functors' from collared cospans to spans of sets, providing - by linearisation - topological quantum field theories (TQFT) on manifolds and their cobordisms. Similarly, (co)homology and homotopy functors take collared cospans to relations of abelian groups or (co) spans of groups, yielding other 'algebraic' invariants. This is the second paper in a series devoted to the study of cospans in Algebraic Topology. It is practically independent from the first, which deals with higher cubical cospans in abstract categories. The third article will proceed from both, studying cubical topological cospans and their collared version. ____________ Marco Grandis