An analogue of the Thom space for maps with a singularity of type $\Sigma^1$

In this paper there is constructed a space $X(k)$, the $(n+k)$th homotopy group of which is interpreted as the cobordism group of $S$-maps of $n$-dimensional manifolds into $\mathbf R^{n+k}$.
From the construction of $X(k)$ and its analogues we derive corollaries concerning the cobordism group of immersions in the metastable range of dimensions.
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