The \(K\)-theory of the cotangent sphere bundle of \(\mathbb{R}P^n\). (English) Zbl 0544.55007
Let \(W_n\) denote the cotangent sphere bundle of \(\mathbb{R}P^n\). The author calculates the following unitary \(K\)-cohomology groups.
Theorem:
\[K^t({\mathbb{R}}P^n)\to KP^t(W_n)\text{ is one-one}; \tag{i}\]
\[\tilde K^0(W_{2n+1})\cong(\mathbb{Z}/2^n)\oplus(\mathbb{Z}/2^n)\oplus \mathbb{Z}, \quad \tilde K^1(W_{2n+1})\cong {\mathbb{Z}}\oplus {\mathbb{Z}}; \tag{ii}\]
\[\tilde K^0(W_{2n})\cong {\mathbb{Z}}/4\text{ if }n=1, \quad \cong(\mathbb{Z}/2^n)\oplus(\mathbb{Z}/2^n)\text{ if }n>1, \quad \tilde K^1(W_{2n})\cong\mathbb{Z}. \tag{iii}\]
The computations are interesting in view of Gilkey’s homomorphism \[ \eta: \tilde K^0(W_{2n+\varepsilon})\to\mathbb{Z}/2^n\quad(\varepsilon = 0,1)\] constructed from the Atiyah-Singer eta-invariant. The theorem shows that \(\eta\) is generally split onto.
Theorem:
\[K^t({\mathbb{R}}P^n)\to KP^t(W_n)\text{ is one-one}; \tag{i}\]
\[\tilde K^0(W_{2n+1})\cong(\mathbb{Z}/2^n)\oplus(\mathbb{Z}/2^n)\oplus \mathbb{Z}, \quad \tilde K^1(W_{2n+1})\cong {\mathbb{Z}}\oplus {\mathbb{Z}}; \tag{ii}\]
\[\tilde K^0(W_{2n})\cong {\mathbb{Z}}/4\text{ if }n=1, \quad \cong(\mathbb{Z}/2^n)\oplus(\mathbb{Z}/2^n)\text{ if }n>1, \quad \tilde K^1(W_{2n})\cong\mathbb{Z}. \tag{iii}\]
The computations are interesting in view of Gilkey’s homomorphism \[ \eta: \tilde K^0(W_{2n+\varepsilon})\to\mathbb{Z}/2^n\quad(\varepsilon = 0,1)\] constructed from the Atiyah-Singer eta-invariant. The theorem shows that \(\eta\) is generally split onto.
Reviewer: Victor Snaith
MSC:
| 55N15 | Topological \(K\)-theory |
| 55Q45 | Stable homotopy of spheres |
| 55R50 | Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory |
| 55Q50 | \(J\)-morphism |
