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Degrees of maps between isotropic Grassmann manifolds. (English) Zbl 1386.55006
Let \(\widetilde G_{n,k}\) be the Grassmann manifold of oriented \(k\)-dimensional subspaces in \(\mathbb R^n\) (\(1\leq k\leq n-k\)). It is well known that \(\widetilde G_{n,k}\) is a closed orientable \(k(n-k)\)-dimensional manifold. It was shown in [V. Ramani and P. Sankaran, Proc. Indian Acad. Sci., Math. Sci. 107, No. 1, 13–19 (1997; Zbl 0884.55002)] that if \((n,k)\neq(m,l)\), \(l\geq2\) and \(k(n-k)=l(m-l)\), then every map \(f:\widetilde G_{n,k}\rightarrow\widetilde G_{m,l}\) has degree zero.
In the paper under review, the authors consider the Grassmann manifold \(\widetilde I_{2n,k}\) (where \(1\leq k\leq n\)) of oriented isotropic \(k\)-dimensional subspaces of \(\mathbb R^{2n}\) (which is equipped with the standard symplectic form) and obtain an analogous result: if \(\widetilde I_{2n,k}\) and \(\widetilde I_{2m,l}\) are two distinct “oriented isotropic Grassmannians” of the same dimension, and if \(k,l\geq2\), then \(\deg f=0\) for all maps \(f:\widetilde I_{2n,k}\rightarrow\widetilde I_{2m,l}\). Moreover, they establish that the same conclusion holds for all maps of the form \(\widetilde I_{2n,k}\rightarrow\widetilde G_{m,l}\) and \(\widetilde G_{m,l}\rightarrow\widetilde I_{2n,k}\), provided that \(\dim\widetilde I_{2n,k}=\dim\widetilde G_{m,l}\) and \(k,l\geq2\). The authors actually prove that there is no ring monomorphism between rational cohomology rings of the manifolds in question and use the following fact (easily obtained from PoincarĂ© duality): if \(f:M\rightarrow N\) is a nonzero degree map between oriented closed connected manifolds of the same dimension, then \(f^*:H^*(N;\mathbb Q)\rightarrow H^*(M;\mathbb Q)\) is a monomorphism.
MSC:
55M25 Degree, winding number
14M17 Homogeneous spaces and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
57T15 Homology and cohomology of homogeneous spaces of Lie groups
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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