On the Components of the Push-out Space with Certain Indices

Abstract

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 in [3]. It is known that the push-out space has finitely many path-connected components and each path-connected component can be assigned an integer called the index of the component. In this study, when $M$ is compact, we give 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.