Geodesic
The degrees of maps between $(n-1)$-connected $(2n+1)$-dimensional manifolds or Poincaré complexes and their applications
Sbornik. Mathematics, Tome 212 (2021) no. 10, pp. 1360-1414 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, using homotopy theoretical methods we study the degrees of maps between $(n-1)$-connected $(2n+1)$-dimensional Poincaré complexes. Necessary and sufficient algebraic conditions for the existence of mapping degrees between such Poincaré complexes are established. These conditions allow us, up to homotopy, to construct explicitly all maps with a given degree. As an application of mapping degrees, we consider maps between ${(n-1)}$-connected $(2n+1)$-dimensional Poincaré complexes with degree $\pm 1$, and give a sufficient condition for these to be homotopy equivalences. This resolves a homotopy theoretical analogue of Novikov's question: when is a map of degree $1$ between manifolds a homeomorphism? For low $n$, we classify, up to homotopy, torsion free $(n-1)$-connected $(2n+1)$-dimensional Poincaré complexes. Bibliography: 29 titles.
Keywords: mapping degree, highly connected manifolds and Poincaré complexes, homotopy theory
Mots-clés : classification of Poincaré complexes. 
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J. Grbić; A. Vučić. The degrees of maps between $(n-1)$-connected $(2n+1)$-dimensional manifolds or Poincaré complexes and their applications. Sbornik. Mathematics, Tome 212 (2021) no. 10, pp. 1360-1414. http://geodesic.mathdoc.fr/item/SM_2021_212_10_a1/

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