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0. Introduction. The notion of a T -structure on a manifold is a generalization of an action of a torus. Tori, possibly of different dimensions, act on open subsets of the manifold and these actions fit together on overlaps in such a way that the torus acting on one of the sets injects homomorphically into the torus acting on the second one. The T -structure was introduced in [G]. More general notions, namely an F -structure and a nilpotent Killing structure, were introduced in [CG1], [CG2] and [CFG]. Here we assume that the manifold is compact, orientable and differentiable and that the local actions of tori are smooth. There is an extra notion of polarization which can be associated with a manifold with a T -structure (or an F -structure). For each open chart Uα there is a choice of a nontrivial linear subspace of the Lie algebra of the acting torus such that the derivative of the action sends it isomorphically onto a linear subspace of the tangent space of Uα at a given point. The bordism groups of oriented 3and 4-manifolds with T -structure and polarization were calculated in [HJ] and [Mi2]. These abelian groups have an uncountable set of generators. A cofinite subset of the Z-basis is parametrized by the slope coefficient of polarizations near singular orbits in strata of codimension 2. The slope coefficient corresponds to an element of the interval (0, 1/2]. A similar situation occurs in dimension 5, where the slope coefficient is calculated near isolated singular orbits of codimension 4. Essential information is encoded in generators of type (11) in Section 5. Let us denote the bordism group of compact oriented n-manifolds with T -structure and polarization by Ω n . In this paper we obtain the following result.

Bordism of oriented 5-manifolds with T-structure and polarization