Osaka Journal of Mathematics

Degrees of maps between Grassmann manifolds

Parameswaran Sankaran and Swagata Sarkar

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Let $f\colon \mathbb{G}_{n,k}\to \mathbb{G}_{m,l}$ be any continuous map between two distinct complex (resp. quaternionic) Grassmann manifolds of the same dimension. We show that the degree of $f$ is zero provided $n,m$ are sufficiently large and $l\geq 2$. If the degree of $f$ is $\pm 1$, we show that $(m,l)=(n,k)$ and $f$ is a homotopy equivalence. Also, we prove that the image under $f^{*}$ of every element of a set of algebra generators of $H^{*}(\mathbb{G}_{m,l};\mathbb{Q})$ is determined up to a sign, $\pm$, by the degree of $f$, provided this degree is non-zero.

Article information

Osaka J. Math., Volume 46, Number 4 (2009), 1143-1161.

First available in Project Euclid: 15 December 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M25: Degree, winding number
Secondary: 57R20: Characteristic classes and numbers 57T15: Homology and cohomology of homogeneous spaces of Lie groups


Sankaran, Parameswaran; Sarkar, Swagata. Degrees of maps between Grassmann manifolds. Osaka J. Math. 46 (2009), no. 4, 1143--1161.

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