Celebratio Mathematica

I. James and E. Thomas: “Which Lie groups are homotopy-abelian?,” Proc. Nat. Acad. Sci. U.S.A. 45 : 5 (May 1959), pp. 737–740. MR 0106465 Zbl 0085.25801 article

S. Araki, I. M. James, and E. Thomas: “Homotopy-abelian Lie groups,” Bull. Am. Math. Soc. 66 : 4 (1960), pp. 324–326. MR 0119207 Zbl 0152.01103 article

Ioan M. James	Related
S. Araki	Related

I. James and E. Thomas: “Homotopy-abelian topological groups,” Topology 1 : 3 (1962), pp. 237–240. MR 0149483 Zbl 0107.40602 article

I. James and E. Thomas: “Homotopy-commutativity in rotation groups,” Topology 1 : 2 (1962), pp. 121–124. MR 0139167 Zbl 0114.39204 article

Let \( R_q \) denote the rotation group in Euclidean \( q \)-space, where \( q\geq 1 \). We make the usual embeddings \[ R_1 \subset R_2 \subset \cdots \subset R_q \cdots\,, \] so that the \( q \)-sphere \( S^q \) is the factor space of \( R_{q+1} \) by \( R_q \). We recall that \( R_q \) is a retract of \( R_{q+1} \) for \( q=1,3,7 \).

Let \( m,n > 1 \). If the commutator map \[ R_m \times R_n \to R_q \qquad (q\geq m,n) \] is nul-homotopic we say that \( R_m \) and \( R_n \) homotopy-commute in \( R_q \). This is true when \( q\geq m+n \), since then \( R_n \) is conjugate in \( R_q \) to a sub-group whose elements commute with those of \( R_m \). We shall describe the pair \( (m,n) \) as irregular if \( R_m \) and \( R_n \) homotopy-commute in \( R_{m+n-1} \), as regular if they do not. The pair \( (m,n) \) is irregular if \( m+n = 4 \) or 8, since then \( R_{m+n-1} \) is a retract of \( R_{m+n} \). Are there any other irregular pairs? In this note we prove

Let \( m+n \neq 4,\,8 \). Then \( (m,n) \) is regular if \( m \) or \( n \) is even or if \( d(m) = d(n) \), where \( d(q) \), for \( q\geq 2 \), denotes the greatest power of 2 which divides \( q-1 \).

Thus the answer to our question is negative for \( m+n < 12 \). The least pairs where the answer is unknown are \( (3,9) \) and \( (5,7) \).

I. James and E. Thomas: “On homotopy-commutativity,” Ann. Math. (2) 76 : 1 (July 1962), pp. 9–17. MR 0139166 Zbl 0107.40601 article

I. M. James, E. Thomas, H. Toda, and G. W. Whitehead: “On the symmetric square of a sphere,” J. Math. Mech. 12 : 5 (1963), pp. 771–776. MR 0154282 Zbl 0136.20301 article

Ioan M. James	Related
Hiro Y. Toda	Related
George W. Whitehead, Jr.	Related

I. James and E. Thomas: “An approach to the enumeration problem for non-stable vector bundles,” J. Math. Mech. 14 : 3 (1965), pp. 485–506. MR 0175134 Zbl 0142.40701 article

Consider real vector bundles over a complex \( A \). (By a complex we always mean a pathwise connected \( CW \)-complex.) How many classes of \( q \)-plane bundles (\( q=0,1,\dots \)) are there which are stably equivalent to a given stable bundle? There may be none if \( q < \dim A \). There is at least one if \( q \geq \dim A \) and precisely one if \( q > \dim A \). We prove theorems which determine the number when \( q = \dim A \), except that we fail to settle the question for non-orientable bundles when \( q \) is even. Our approach is suggested, in part, by theorems of Eilenberg and Mac Lane [1954] on spaces with two non-vanishing homotopy groups. We study the properties of pairs of spaces which have one non-vanishing relative homotopy group, up to a certain dimension. Certain pairs of classifying spaces satisfy the right conditions and hence our results on vector bundles are obtained.

I. James and E. Thomas: “Note on the classification of cross-sections,” Topology 4 : 4 (January 1966), pp. 351–359. MR 0212820 Zbl 0136.44202 article

I. James and E. Thomas: “On the enumeration of cross-sections,” Topology 5 : 2 (May 1966), pp. 95–114. MR 0196753 Zbl 0141.40505 article

I. James and E. Thomas: “Submersions and immersions of manifolds,” Invent. Math. 2 : 3 (1967), pp. 171–177. MR 0212821 Zbl 0146.19803 article

E. Thomas and A. T. Vasquez: “Betti numbers of Hilbert modular varieties,” pp. 70–87 in Topological topics: Articles on algebra and topology presented to P. J. Hilton in celebration of his sixtieth birthday. Edited by I. M. James. London Mathematical Society Lecture Note Series 86. Cambridge University Press, 1983. MR 827249 Zbl 0523.14026 incollection

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Alphonse Thomas Vasquez	Related
Peter John Hilton	Related

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