On the components of the push-out space with certain indices (Q442622)

Given an immersion of a connected, \(m\)-dimensional manifold \(M\) without boundary into the Euclidean \((m + k)\)-dimensional space, the idea of the push-out space of the immersion under the assumption that immersion has flat normal bundle is introduced by \textit{S. Carten} and \textit{Z. Sentürk} in [J. Lond. Math. Soc., II. Ser. 50, No. 2, 404--416 (1994; Zbl 0807.53042)]. It is known that the push-out space has finitely many path-connected components and to each path-connected component can be assigned an integer called the index of the component. In the present work, for compact \(M\), the author gives some new results on the push-out space. Especially it is proved that if the push-out space has a component with index 1, then the Euler number of \(M\) is 0 and if the immersion has a co-dimension 2, then the number of path-connected components of the push-out space with index \((m - 1)\) is at most 2.