## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 43, Number 4 (2003), 815-827.

### Adams $e$-invariant, Toda bracket and $[X, U(n)]$

#### Abstract

In the previous paper [1], the author investigated the group structure of the homotopy set $[X,U(n)]$ with the pointwise multiplication, under the assumption that $X$ is a finite CW complex with its dimension $2n$ and $U(n)$ is the unitary group, and showed that $[X,U(n)]$ is an extension of $\Tilde{K}^{1}(X)$ by $N_{n}(X)$, where $N_{n}(X)$ is a group defined as the cokernel of a map $\Theta : \Tilde{K}^{0}(X)\to \mathrm{H}^{2n}(X;\mathbf{Z})$. In this paper, we offer another interpretation of $N_{n}(X)$ using Adams $e$-invariant and show that the extension $N_{n}(X) \to U_{n}(X) \to \Tilde{K}^{1}(X)$ is determined by some Toda brackets. Also we give some applications including the calculation of $[SO(4), U(3)]$.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 43, Number 4 (2003), 815-827.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250281737

**Digital Object Identifier**

doi:10.1215/kjm/1250281737

**Mathematical Reviews number (MathSciNet)**

MR2030800

**Zentralblatt MATH identifier**

1062.55012

**Subjects**

Primary: 55Q35: Operations in homotopy groups

#### Citation

Hamanaka, Hiroaki. Adams $e$-invariant, Toda bracket and $[X, U(n)]$. J. Math. Kyoto Univ. 43 (2003), no. 4, 815--827. doi:10.1215/kjm/1250281737. https://projecteuclid.org/euclid.kjm/1250281737