Geodesic
Antipodes and embeddings
Sbornik. Mathematics, Tome 196 (2005) no. 1, pp. 1-28

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This paper is concerned with maps without antipodal coincidence from spheres into compacta and polyhedra of a smaller dimension and to obstructions for embeddings of polyhedra and compacta in Euclidean spaces. Estimates of the dimension of the antipodal coincidence set are given for maps of spheres into compacta. The theory of the Yang homology index of spaces with involution is systematically expounded and developed in the case of a deleted square. 
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A. Yu. Volovikov; E. V. Shchepin. Antipodes and embeddings. Sbornik. Mathematics, Tome 196 (2005) no. 1, pp. 1-28. http://geodesic.mathdoc.fr/item/SM_2005_196_1_a0/