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A note on nontrivial intersection for selfmaps of complex Grassmann manifolds. (English) Zbl 1387.55003
The setting of this note is provided by the complex Grassmann manifold \(\mathbb{C}M(k, n)=G(k, n)\), the space of \(k\)-planes in \(\mathbb{C}^{n+k}\). The main result (Theorem 4) states that for every continuous selfmap \(f:G(k, n)\rightarrow G(k, n)\) there exists \(V^k\in G(k, n)\) such that \(V^k\cap f(V^k)\neq \{0\}\).
The main tool in the proof is the total Chern class of \(\gamma ^k\), the canonical \(k\)-plane bundle over \(G(k, n)\).
MSC:
55M20 Fixed points and coincidences in algebraic topology
57T15 Homology and cohomology of homogeneous spaces of Lie groups
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Full Text: Euclid