## Osaka Journal of Mathematics

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

### Degrees of maps between Grassmann manifolds

Parameswaran Sankaran and Swagata Sarkar

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 15 December 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ojm/1260892843

**Mathematical Reviews number (MathSciNet)**

MR2604924

**Zentralblatt MATH identifier**

1185.55003

**Subjects**

Primary: 55M25: Degree, winding number

Secondary: 57R20: Characteristic classes and numbers 57T15: Homology and cohomology of homogeneous spaces of Lie groups

#### Citation

Sankaran, Parameswaran; Sarkar, Swagata. Degrees of maps between Grassmann manifolds. Osaka J. Math. 46 (2009), no. 4, 1143--1161. https://projecteuclid.org/euclid.ojm/1260892843