A note on homotopy types of connected components of Map \((S^4, BSU(2))\) (Q425285)
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scientific article; zbMATH DE number 6043587
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A note on homotopy types of connected components of Map \((S^4, BSU(2))\) |
scientific article; zbMATH DE number 6043587 |
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A note on homotopy types of connected components of Map \((S^4, BSU(2))\) (English)
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8 June 2012
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principal \(G\) bundle
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homotopy type
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bundle map
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gauge group
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classifying space
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\(A_n\)-type
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connected components
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It is known that the \(k\)-th connected component of the mapping space \({\text Map}(S^4, BSU(2))\) is homotopy equivalent to the classifying space of the gauge group \({\mathcal G}_k\) of the principal \(SU(2)\)-bundle over \(S^4\) with its second Chern number \(k\). \textit{S. Tsukuda} [J. Pure Appl. Algebra 161, No.1--2, 235--243 (2001; Zbl 0976.55004)] obtained the \(p\)-local homotopy type of its connected components, but his proof was not complete for the case \(p=2\).NEWLINENEWLINE In this paper, the author gives the complete proof by using the Chern character and some numerical computations of a sequence \(\{\epsilon_i\}_{i=0}^{\infty}\) of rational numbers. Moreover, he also studies the \(A_n\)-type of the gauge group \({\mathcal G}_k\) by using the \(\epsilon_i\)'s and obtains some upper bound for \(A_n\) types.
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