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# Homology, Homotopy and Applications

## Volume 25 (2023)

### Number 2

### $K$-theory of real Grassmann manifolds

Pages: 383 – 402

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n2.a17

#### Authors

#### Abstract

Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb{R}^n$. We compute the complex $K$-ring of $G_{n,k}\:$, up to a small indeterminacy, for all values of $n,k$ where $2 \leqslant k \leqslant n - 2$. When $n \equiv 0 (\operatorname{mod} 4), k \equiv 1 (\operatorname{mod} 2)$, we use the Hodgkin spectral sequence to determine the $K$-ring completely.

#### Keywords

real Grassmann manifold, $K$-theory, Hodgkin spectral sequence

#### 2010 Mathematics Subject Classification

19L99, 55N15

Copyright © 2023, Sudeep Podder and Parameswaran Sankaran. Permission to copy for private use granted.

Received 23 April 2022

Received revised 27 November 2022

Accepted 8 December 2022

Published 22 November 2023