Geodesic
Subvarieties of codimensionality one and of simple homotopic type
Matematičeskie zametki, Tome 12 (1972) no. 2, pp. 167-175.

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Let $f_1: M_1\to M_2$ be a homotopic equivalence and $N\subset M_2$ a subvariety of codimensionality one. In this paper we indicate the conditions under which there is a mapping $f_2\sim f_1$ such that the variety $f_2^{-1}(N)$ is simply homotopically equivalent to the variety $N$. 
@article{MZM_1972_12_2_a8,
     author = {S. B. Shlosman},
     title = {Subvarieties of codimensionality one and of simple homotopic type},
     journal = {Matemati\v{c}eskie zametki},
     pages = {167--175},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_12_2_a8/}
}
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S. B. Shlosman. Subvarieties of codimensionality one and of simple homotopic type. Matematičeskie zametki, Tome 12 (1972) no. 2, pp. 167-175. http://geodesic.mathdoc.fr/item/MZM_1972_12_2_a8/