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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
##### Keywords:
fixed point; complex Grassmann manifold
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