Celebratio Mathematica

[1] phdthesis E. Thomas: A generalization of the Pontrjagin square cohomology operation. Ph.D. thesis, Princeton University, 1955. Advised by N. E. Steenrod. Published (under a different title) as Memoirs of the American Mathematical Society 27 (1957). See also Proc. Nat. Acad. Sci. U.S.A. 42:5 (1956).

[2]E. Thomas: “A generalization of the Pontrjagin square cohomology operation,” Proc. Nat. Acad. Sci. U.S.A. 42 : 5 (May 1956), pp. 266–269. See also the author’s PhD thesis (1955). MR 0079254 Zbl 0071.16302 article

[3]N. E. Steenrod and E. Thomas: “Cohomology operations derived from cyclic groups,” Comment. Math. Helv. 32 (1957), pp. 129–152. MR 0092148 Zbl 0090.39101 article

In a previous paper [1957], Steenrod defined a family of cohomology operations, called reduced powers, each being associated with some permutation group. It was also shown that these operations have a basis, in the sense of composition, consisting of, firstly, four primitive types of operations (which are: addition, cup-product, homomorphisms induced by coefficient homomorphisms, and Bockstein–Whitney coboundary operators) and, secondly, those reduced powers associated with cyclic permutation groups having degree \( p \) and order \( p \) where \( p \) ranges over primes.

In this paper, we shall improve the result by showing that there is a smaller basis consisting of the same primitive operations and only particular operations associated with cyclic groups: namely, for each prime \( p \) the cyclic reduced powers \[ \mathcal{P}^i: H^q(K;Z_p) \to H^{q+2i(p-1)}(K;Z_p),\qquad i = 0, 1, \dots , \] and the Pontrjagin \( p \)-th powers \[ \mathfrak{P}_p: H^{2q}(K;Z_{p^k}) \to H^{2pq}(K; Z_{p^{k+1}}). \] The latter were defined for \( p = 2 \) by Pontrjagin [1942], and generalized for \( p > 2 \) by Thomas [1956]. When \( p = 2 \), \( \mathcal{P}^i \) is usually written \( \operatorname{Sq}^{2i} \).

[4]E. Thomas: The generalized Pontrjagin cohomology operations and rings with divided powers. Memoirs of the American Mathematical Society 27. 1957. Republication (under a different title) of the author’s PhD thesis (1955). MR 0099029 Zbl 0085.37504 book

[5]F. P. Peterson and E. Thomas: “A note on non-stable cohomology operations,” Bol. Soc. Mat. Mex., II. Ser. 3 (1958), pp. 13–18. MR 0105680 Zbl 0121.39604 article

[6]E. Thomas: “The generalized Pontrjagin cohomology operations,” pp. 155–158 in Symposium internacional de topología algebraica [International symposium on algebraic topology] (Mexico City, August 1956). Universidad Nacional Autónoma de México and UNESCO (Mexico City and Paris), 1958. MR 0099028 Zbl 0123.16104 incollection

[7]E. Thomas: “The functional Pontrjagin cohomology operations,” Bol. Soc. Mat. Mex., II. Ser. 3 (1958), pp. 19–24. MR 0103460 Zbl 0121.39701 article

[8]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

[9]E. Thomas: “On tensor products of \( n \)-plane bundles,” Arch. Math. Logik Grundlagenforsch 10 : 1 (1959), pp. 174–179. MR 0107234 Zbl 0192.29501 article

[10]E. Thomas: “The suspension of the generalized Pontrjagin cohomology operations,” Pac. J. Math. 9 : 3 (July 1959), pp. 897–911. MR 0110098 Zbl 0121.39603 article

[11]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

[12]E. Thomas: “A note on certain polynomial algebras,” Proc. Am. Math. Soc. 11 : 3 (1960), pp. 410–414. MR 0121393 Zbl 0098.36201 article

[13]E. Thomas: “On the cohomology of the real Grassmann complexes and the characteristic classes of \( n \)-plane bundles,” Trans. Am. Math. Soc. 96 : 1 (July 1960), pp. 67–89. MR 0121800 Zbl 0098.36301 article

[14]E. Thomas: “On functional cup-products and the transgression operator,” Arch. Math. 12 : 1 (1961), pp. 435–444. MR 0149485 Zbl 0101.15705 article

Let \( g \) be a map from a topological space \( X \) to a topological space \( Y \). Take cohomology groups with coefficients in a fixed ring \( R \) and denote by \( g^* \) the homomorphism from \( H^*(Y) \) to \( H^*(X) \) induced by \( g \). Suppose that \( u \in H^{p+1}(Y) \) and \( v\in H^{q+1}(Y) \) are elements such that \( u \smile v = 0 \) and either \( g^*u = 0 \) or \( g^*v = 0 \). Then Steenrod [1949] has defined a functional cup-product \[ u \mathbin{\mathop{\smile}\limits_{g}} v \in H^{p+q+1}(X) \mod L(g,u,v), \] where \[ L(g,u,v) = g^* H^{p+q+1}(Y) + g^*u \smile H^q(X) + H^p(X) \smile g^*v. \] Suppose now that \( X \) and \( Y \) are the total spaces of respective proper triads and that \( g \) is a triad map. The purpose of this paper is to show that it is then possible, in certain cases, to express the functional cup-product in terms of an ordinary cup-product. We apply this to obtain a classical result about the Hopf invariant and to obtain some new information about the cohomology rings of certain classifying spaces.

[15]W. Browder and E. Thomas: “Axioms for the generalized Pontryagin cohomology operations,” Q. J. Math., Oxf. II. Ser. 13 : 1 (1962), pp. 55–60. MR 0140103 Zbl 0104.39701 article

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

[17]E. Thomas: “The torsion Pontryagin classes,” Proc. Am. Math. Soc. 13 : 3 (1962), pp. 485–488. MR 0141132 Zbl 0109.16001 article

[18]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 \( (5,7) \).

[19]E. Thomas: “On the cohomology groups of the classifying space for the stable spinor groups,” Bol. Soc. Mat. Mex., II. Ser. 7 (1962), pp. 57–69. MR 0153027 Zbl 0124.16401 article

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

[21]E. Thomas: “On the mod 2 cohomology of certain \( H \)-spaces,” Comment. Math. Helv. 37 (1962–1963), pp. 132–140. MR 0149484 Zbl 0108.35705 article

Let \( X \) be a topological space and \( G \) an abelian group. We denote by \( H^i(X;G) \) the \( i \)-th (singular) cohomology group of \( X \) with coefficients in \( G \). Define an integer-valued function \( v_G \) by \[ v_G(X) = \text{least positive integer } n \text{ such that } H^n(X;G) \neq 0 \] In case \( X \) is \( G \)-acyclic we set \( v_G(X)=0 \). For simplicity we write the function as \( v_r \) if \( G=Z_r \), the integers mod \( r \), \( r\geq 2 \).

Suppose now that \( X \) is an \( H \)-space–that is, \( X \) has a continuous multiplication with unit–and suppose (for the remainder of the paper) that \( X \) satisfies the following conditions.

The integral cohomology groups of \( X \) are finitely generated in each dimension.
\( H^*(X;Z_2) \) is finitely generated as a vector space, and is primitively generated as a Hopf algebra.

In [1963] we showed that for such \( H \)-spaces, \( v_2(X) = 2^r - 1 \) for some \( r\geq 0 \). The purpose of this note is to prove

Let \( X \) be an \( H \)-space satisfying the above two conditions. Then \[ v_2(X) = 0,\,1,\,3,\,7 \textrm { or } 15. \] Moreover if \( X \) has no 2-torsion, then \[ v_2(X) = v_Q(X) = 0,\,1,\,3 \textrm { or } 7, \] where \( Q \) denotes the rational numbers.

[22]E. Thomas: “Steenrod squares and \( H \)-spaces,” Ann. Math. (2) 77 : 2 (March 1963), pp. 306–317. MR 0145526 Zbl 0115.40303 article

Let \( X \) be a space which has a continuous multiplication with unit; that is, \( X \) is an \( H \)-space. If the integral cohomology groups of \( X \) are of finite type, then the results of Hopf [1941] and Borel [1953] give a general description of the cohomology ring of \( X \) with a field for coefficients. Moreover, if \( X \) is a classical Lie group, and if we take the field of coefficients to be the integers mod a prime number \( p \), then the cohomology of \( X \) is completely known as an algebra over the mod \( p \) Steenrod algebra. Apart from these Lie groups, not a great deal is known about the behavior of the Steenrod operations in the mod \( p \) cohomology of \( H \)-spaces. The purpose of this paper is to obtain some information about this problem, for the special case of the Steenrod squares, \( \operatorname{Sq}^i \).

[23]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

[24]W. Browder and E. Thomas: “On the projective plane of an \( H \)-space,” Ill. J. Math. 7 : 3 (1963), pp. 492–502. MR 0151974 Zbl 0136.43905 article

Let \( G \) be a topological group. Milnor [1956] defines a sequence of principal \( G \)-bundles \( (E_n,B_n,G) \) (\( 1 \leq n < \infty \)) such that \[ SG = B_1 \subset B_2 \subset \cdots \subset B_{\infty} = B_G, \] where \( SG \) denotes the suspension of \( G \) and \( B_G \) is a classifying space for \( G \). The work of Borel [1953; 1954] gives relations between the cohomology of \( G \) and that of \( B_G \), whereas Rothenberg [1961] investigates the cohomology of the spaces \( B_n \).

Suppose now that \( X \) is an \( H \)-space, that is, \( X \) has a continuous multiplication with unit. One then may not be able to define a classifying space \( B_X \), but Stasheff [1961] has defined the projective plane of \( X \), \( P_2X \), which has the homotopy type of the space \( B_2 \) in case \( X \) is actually a group.The purpose of this paper is to discuss the relationship between the cohomology of \( X \) and that of \( P_2X \).

[25]E. Thomas: “Exceptional Lie groups and Steenrod squares,” Mich. Math. J. 11 : 2 (1964), pp. 151–156. MR 0163307 Zbl 0136.44001 article

[26]E. Thomas: “Homotopy classification of maps by cohomology homomorphisms,” Trans. Am. Math. Soc. 111 : 1 (1964), pp. 138–151. MR 0160212 Zbl 0119.18401 article

[27]E. Thomas: “On cross sections to fiber spaces,” Proc. Nat. Acad. Sci. U.S.A. 54 : 1 (July 1965), pp. 40–41. MR 0178476 Zbl 0132.19701 article

[28]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.

[29]E. Thomas: “Steenrod squares and \( H \)-spaces, II,” Ann. Math. (2) 81 : 3 (May 1965), pp. 473–495. MR 0177408 Zbl 0127.13503 article

[30]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

[31]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

[32]E. Thomas: “The Steenrod squares in the mod two cohomology algebra of an \( H \)-space,” pp. 113–117 in Colloque de topologie [Topology symposium] (Brussels, 7–10 September 1964). Publications du CBRM 22. Centre Belge de Recherches Mathématiques (Liège), 1966. MR 0220276 Zbl 0136.43904 incollection

[33]E. Thomas: Seminar on fiber spaces (Berkeley, 1964; Zürich, 1965). Lecture Notes in Mathematics 13. Springer (Berlin), 1966. MR 0203733 Zbl 0151.31604 book

[34]E. Thomas: “Cross-sections of stably equivalent vector bundles,” Q. J. Math., Oxf. II. Ser. 17 : 1 (1966), pp. 53–57. MR 0196747 Zbl 0135.41301 article

[35]E. Thomas: “An exact sequence for principal fibrations,” Bol. Soc. Mat. Mex., II. Ser. 12 (1967), pp. 35–45. MR 0243526 Zbl 0183.51702 article

[36]E. Thomas: “Fields of tangent \( k \)-planes on manifolds,” Invent. Math. 3 : 4 (1967), pp. 334–347. MR 0217814 Zbl 0162.55402 article

Let \( M \) be a manifold, assumed throughouit the paper to be smooth, closed, and connected. We consider the question of whether \( M \) has a smooth field of tangent \( k \)-planes, \( 1 < k < \dim M \). Such a field corresponds to a \( k \)-plane sub-bundle in the tangent bundle of \( M \). We call such a bundle a \( k \)-distribution, following Chevalley [1946]. We will assume that our manifolds are orientable, and we restrict attention to \( k \)-distributions that are oriented bundles.

In a previous paper [1967] we studied the problem of 2-distributions on even-dimensional manifolds. In this paper, we handle the odd-dimensional case. Also, we obtain results on the existence of 3-distributions on manifolds with dimension \( \equiv 2 \mod 4 \). Finally, we compute the first obstruction to a manifold \( M \) having a \( k \)-distribution for \( 1 < k < \dim M \).

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

[38]E. Thomas: “The index of a tangent 2-field,” Comment. Math. Helv. 42 (1967), pp. 86–110. Dedicated to Professor H. Hopf. MR 0215317 Zbl 0153.53504 article

[39]E. Thomas: “Complex structures on real vector bundles,” Am. J. Math. 89 : 4 (October 1967), pp. 887–908. MR 0220310 Zbl 0174.54802 article

[40]E. Thomas: “Postnikov invariants and higher order cohomology operations,” Ann. Math. (2) 85 : 2 (March 1967), pp. 184–217. MR 0210135 Zbl 0152.22002 article

[41]E. Thomas: “Real and complex vector fields on manifolds,” J. Math. Mech. 16 : 11 (1967), pp. 1183–1205. MR 0210136 Zbl 0153.53503 article

[42]E. Thomas: “Fields of tangent 2-planes on even-dimensional manifolds,” Ann. Math. (2) 86 : 2 (September 1967), pp. 349–361. MR 0212834 Zbl 0168.21401 article

We consider in this paper the problem of whether a smooth manifold \( M \) admits a field of tangent 2-planes. If \( M \) has two linearly independent tangent vector fields, these span a field of 2-planes; but we will see that, even if \( M \) has no non-singular vector field, it still may have a field of 2-planes. We consider only the case of even-dimensional manifolds. The odd-dimensional case will be treated subsequently [Thomas 1967]. Of course, if \( \dim M = 2 \), a field of 2-planes exists trivially. Furthermore, Hirzebruch and Hopf [1958] have solved completely the problem of fields of 2-planes on 4-dimensional manifolds; and so, for the rest of the paper, we assume that \( \dim M > 4 \).

[43]E. Thomas: “Submersions and immersions with codimension one or two,” Proc. Am. Math. Soc. 19 : 4 (1968), pp. 859–863. MR 0229246 Zbl 0169.26102 article

Let \( M \) and \( N \) be smooth manifolds and \( f: M\to N \) a smooth map. We say that \( f \) has maximal rank if at each point of \( M \) the Jacobian matrix of \( f \) has maximal rank. If \( \dim M < \dim N \), then \( f \) is an immersion; while if \( \dim M > \dim N \), \( f \) is a submersion. For convenience we define the integer \( |\dim M - \dim N| \) to be the codimension of any map \( M\to N \). We consider in this note the following problem. Let \( g:M\to N \) be a continuous map of codimension one or two. When is \( g \) homotopic to a smooth map of maximal rank? By exploiting the work of Hirsch [1959] and Phillips [1966] we obtain answers in terms of cohomology invariants of \( M \) and \( N \).

[44]E. Thomas: “Vector fields on low dimensional manifolds,” Math. Z. 103 : 2 (1968), pp. 85–93. MR 0224109 Zbl 0162.55403 article

[45]A. Hughes and E. Thomas: “A note on certain secondary cohomology operations,” Bol. Soc. Mat. Mex., II. Ser. 13 (1968), pp. 1–17. MR 0266203 Zbl 0207.21801 article

[46]E. Thomas: “The span of a manifold,” Q.J. Math., Oxf. II. Ser. 19 : 1 (1968), pp. 225–244. MR 0234487 Zbl 0167.51901 article

[47]D. Frank and E. Thomas: “A generalization of the Steenrod–Whitehead vector field theorem,” Topology 7 : 3 (August 1968), pp. 311–316. MR 0229253 Zbl 0186.57303 article

[48]E. Thomas: “On the existence of immersions and submersions,” Trans. Am. Math. Soc. 132 : 2 (1968), pp. 387–394. MR 0225332 Zbl 0169.26101 article

[49]E. Thomas: “Vector fields on manifolds,” Bull. Am. Math. Soc. 75 (1969), pp. 643–683. MR 0242189 Zbl 0183.51703 article

[50]E. Thomas: “The Thom–Massey approach to embeddings,” pp. 283–307 in The Steenrod algebra and its applications (Battelle Memorial Institute, Columbus, OH, 30 March–4 April 1970). Edited by F. P. Peterson. Lecture Notes in Mathematics 168. Springer (Berlin), 1970. Proceedings of a conference to celebrate N. E. Steenrod’s sixtieth birthday. MR 0287560 Zbl 0216.45302 incollection

Franklin P. Peterson	Related
N. E. Steenrod	Related

[51]E. Thomas: “Whitney–Cartan product formulae,” Math. Z. 118 : 2 (1970), pp. 115–138. MR 0278296 Zbl 0205.53103 article

[52]E. Thomas: “Some remarks on vector fields,” pp. 107–113 in H-spaces: Actes de la réunion de Neuchâtel [H-spaces: Proceedings of the Neuchâtel meeting] (Neuchâtel, Switzerland, August 1970). Edited by F. Sigrist. Lecture Notes in Mathematics 196. Springer (Berlin), 1971. MR 0287567 Zbl 0218.57013 incollection

[53] incollection L. L. Larmore and E. Thomas: “Group extensions and principal fibrations” in Proceedings of the advanced study institute on algebraic topology (Aarhus, Denmark, 10–13 August 1970), vol. 2. Various publications series 13. Aarhus Universitet Matematisk Enstitut, 1971. Also published as Math. Scand. 30 (1972).

[54]L. L. Larmore and E. Thomas: “Group extensions and principal fibrations,” Math. Scand. 30 (1972), pp. 227–248. Also appeared in Proceedings of the advanced study institute on algebraic topology (1970). MR 0328935 Zbl 0254.55009 article

We consider in this paper the following extension problem. Suppose we have a principal fibration \[ \Omega C \stackrel{i}{\longrightarrow} E \stackrel{\pi}{\longrightarrow}B, \] with classifying map \( \theta:B\to C \). One then has an extended sequence of fibrations \[ \cdots \to \Omega^{m+1}C \stackrel{{}^{m}i}{\longrightarrow} \Omega^m E\stackrel{{}^{m}\pi}{\longrightarrow} \Omega^m B \to \cdots \to B \stackrel{\theta}{\longrightarrow} C. \] (For any map \( f \) we write \( ^mf = \Omega^m f \), \( m\geq 1 \).) Given a space \( X \) a standard problem in topology is to compute the group \( [X,\Omega^m E] \). We have the exact sequence \[ e: 0 \to (\operatorname{Coker}^{m+1}\theta_*)\to [X,\Omega^m E] \to (\operatorname{Ker}^m\theta_*) \to 0, \] and our problem is: Compute the extension \( e \).

[55]E. Thomas: “Cohomology operations on \( p \)-fold sums,” Proc. Am. Math. Soc. 33 : 2 (June 1972), pp. 551–553. MR 0290356 Zbl 0242.55024 article

[56]L. L. Larmore and E. Thomas: “Mappings into loop spaces and central group extensions,” Math. Z. 128 : 4 (1972), pp. 277–296. MR 0317318 Zbl 0254.55010 article

[57]E. Thomas: “Secondary obstructions to integrability,” pp. 655–661 in Dynamical systems (Salvador, Brazil, 26 July–14 August 1971). Edited by M. M. Peixoto. Academic Press (New York), 1973. MR 0343293 Zbl 0285.58005 incollection

[58]L. L. Larmore and E. Thomas: “Group extensions and twisted cohomology theories,” Ill. J. Math. 17 : 3 (1973), pp. 397–410. MR 0336744 Zbl 0274.55022 article

In this paper we continue the study of group extensions initiated in [Larmore and Thomas 1972]. The specific problem discussed there was the computation of extensions in the exact sequence of groups obtained by mapping a space into a principal fibration sequence. Here we consider the same problem, but in a different category — the category of spaces “over and under” a fixed space (see [McClendon 1969; Becker 1968]). This means in particular that the solution to the extension problem is given in terms of “twisted” cohomology operations [McClendon 1969], whereas in [Larmore and Thomas 1972] only ordinary cohomology operations were needed.

In §1 we discuss the category we will use. In §2 we state our extension problem, and in §§3–4 we give a general solution. Finally in §§5–6 we give applications of our theory — in §5 we compute the (affine) group of immersions of an \( n \) manifold in \( R^{2n-1} \), while in §6 we compute the (affine) group of vector 1-fields on a manifold.

[59]J. Wood and E. Thomas: “On signatures associated with ramified coverings and embedding problems,” pp. 229–235 in Colloque international sur l’analyse et la topologie différentielle [International symposium on analysis and differential topology] (CNRS, Strasbourg, 20–29 June 1972), published as Ann. Inst. Fourier 23 : 2 (1973). Colloques internationaux du Centre national de la recherche scientifique 210. MR 0339201 Zbl 0262.57012 incollection

[60]E. Thomas and R. S. Zahler: “Nontriviality of the stable homotopy element \( \gamma_1 \),” J. Pure Appl. Algebra 4 : 2 (April 1974), pp. 189–203. MR 0356049 Zbl 0287.55014 article

[61]R. Hartshorne, E. Rees, and E. Thomas: “Nonsmoothing of algebraic cycles on Grassmann varieties,” Bull. Am. Math. Soc. 80 : 5 (September 1974), pp. 847–851. MR 0357402 Zbl 0289.14011 article

By a cycle \( Z \) of dimension \( r \) on a nonsingular algebraic variety \( X \), we mean a formal linear combination \( Z = \sum n_iY_i \) of irreducible subvarieties \( Y_i \) of dimension \( r \), with integer coefficients \( n_i \). The smoothing problem for cycles asks whether a given cycle \( Z \) is equivalent (for a suitable equivalence relation of cycles, such as rational equivalence or algebraic equivalence) to a cycle \( Z^{\prime} = \sum n^{\prime}_iY^{\prime}_i \), where the subvarieties \( Y^{\prime}_i \) are all nonsingular. Let \( X \) be a nonsingular projective variety of dimension \( n \) over \( \mathbf{C} \). Then for each cycle \( Z \) of dimension \( r \) on \( X \) we can assign a cohomology class \[ \delta(Z)\in H^{2n-2r}(X,\mathbf{Z}) .\] We say that two cycles \( Z \), \( Z^{\prime} \) are homologically equivalent if \( \delta(Z) = \delta(Z^{\prime}) \). Our main result is that there are cycles on certain Grassmann varieties which cannot be smoothed, even for homological equivalence, which is weaker than rational or algebraic equivalence.

Elmer Rees	Related
Robin Cope Hartshorne	Related

[62]E. Thomas and J. Wood: “On manifolds representing homology classes in codimension 2,” Invent. Math. 25 : 1 (1974), pp. 63–89. MR 0383438 Zbl 0283.57018 article

[63]E. Thomas: “Integrality relations on smooth manifolds,” Math. Scand. 39 : 2 (1976), pp. 195–231. MR 0442933 Zbl 0356.57006 article

[64]E. Thomas: “Embedding manifolds in Euclidean space,” Osaka J. Math. 13 : 1 (1976), pp. 163–186. MR 0474332 Zbl 0328.57009 article

[65]E. Thomas and R. Zahler: “Generalized higher order cohomology operations and stable homotopy groups of spheres,” Adv. Math. 20 : 3 (June 1976), pp. 287–328. MR 0423351 Zbl 0335.55009 article

[66]E. Rees and E. Thomas: “On the divisibility of certain Chern numbers,” Q. J. Math., Oxf. II. Ser. 28 : 4 (1977), pp. 389–401. MR 0467771 Zbl 0401.55009 article

[67]E. Rees and E. Thomas: “Smoothings of isolated singularities,” pp. 111–117 in Algebraic and geometric topology (Stanford, CA, 2–21 August 1976), part 2. Edited by R. J. Milgram. Proceedings of Symposia in Pure Mathematics 32. American Mathematical Society (Providence, RI), 1978. MR 520527 Zbl 0398.57001 incollection

Elmer Rees	Related
Richard James Milgram	Related

[68]E. Rees and E. Thomas: “Cobordism obstructions to deforming isolated singularities,” Math. Ann. 232 : 1 (1978), pp. 33–53. MR 0500994 Zbl 0381.57011 article

If \( V \) is a projective complex variety with an isolated singularity at the point \( p \), it is natural to ask whether one can deform \( V \) slightly so that it becomes non-singular near \( p \). When \( V \) is a hypersurface (or a complete intersection) it is well known that such a deformation is possible — one alters the defining equations slightly. It is tempting to think that this can always be done, but R. Thom, in unpublished work (now reproduced in [Hartshorne 1974]), gave an example of a variety with an isolated singularity that is not smoothable. Thom’s method was to use cobordism to show that if one lets \( S \) denote the boundary of a small disc \( D \) centered at \( p \) and \( K=S\cap V \), then the manifold \( K \) does not bound a smooth manifold in \( D \). Of course if \( V \) can be deformed slightly to become nonsingular,the manifold \( K \) does bound in \( D \), thus Thom gives a necessary condition for an isolated singularity to be smoothable. In this paper we investigate Thom’s idea carefullly and we find that there are more delicate invariants than his which can nevertheless be calculated.

[69]E. Rees and E. Thomas: “Complex cobordism and intersections of projective varieties,” pp. 168–178 in Variétés analytiques compactes [Compact analytic varieties] (Nice, 19–23 September 1977). Edited by Y. Hervier and A. Hirschowitz. Lecture Notes in Mathematics 683. Springer (Berlin), 1978. MR 517524 Zbl 0427.57016 incollection

Complete intersections are, in many ways, the best understood and simplest projective varieties. So, it is interesting to know the extent to which a given projective variety differs from a complete intersection. An obvious first step is to ask whether the given variety \( X_n\subset P_{n+k} \) is the transverse intersection of a hypersurface \( H \) with some smooth variety \( Y_{n+1}\subset P_{n+k} \). In this paper we use complex cobordism to study this question for smooth complex varieties. We obtain integrality conditions on the various transverse intersections of \( X_n \) with the linear subspaces of \( P_{n+k} \). In the case of high codimension, these conditions can be stated very simply in terms of complex cobordism. When the codimension is lower (\( n\geq k+2 \)) there are some extra, unstable, restrictions which are analogous to those that we have already studied in [Rees and Thomas 1978]; for the case \( n=k+2 \) of our present problem we study them here. The precise relationship between these unstable restrictions and the problem studied in [Rees and Thomas 1978] will be discussed at the end of §3. As well as obtaining these (necessary) integrality conditions we show that they are also sufficient for one to be able to find a “complex normal” submanifold \( Y \) such that \( Y \cap H = X \). In this introduction we have stressed this particular application, however it is a consequence of a much more general result that we discuss in §2.

Elmer Rees	Related
Yves Hervier	Related
Andras Hirschowitz	Related

[70]E. Thomas: “Fundamental units for orders in certain cubic number fields,” J. Reine Angew. Math. 1979 : 310 (1979), pp. 33–55. MR 546663 Zbl 0427.12005 article

[71]E. Thomas and A. T. Vasquez: “On the resolution of cusp singularities and the Shintani decomposition in totally real cubic number fields,” Math. Ann. 247 : 1 (1980), pp. 1–20. MR 565136 Zbl 0403.14005 article

[72]L. L. Larmore and E. Thomas: “On the fundamental group of a space of sections,” Math. Scand. 47 : 2 (1980), pp. 232–246. MR 612697 Zbl 0463.55010 article

[73]E. Rees and E. Thomas: “Realizing homology classes by almost-complex submanifolds,” Math. Z. 172 : 2 (1980), pp. 195–201. MR 580860 Zbl 0443.14010 article

[74]E. Thomas and A. T. Vasquez: “Diophantine equations arising from cubic number fields,” J. Number Theory 13 : 3 (August 1981), pp. 398–414. MR 634208 Zbl 0468.10009 article

[75]E. Thomas and A. T. Vasquez: “Chern numbers of Hilbert modular varieties,” J. Reine Angew. Math. 1981 : 324 (1981), pp. 192–210. MR 614525 Zbl 0491.14019 article

[76]E. Thomas and A. T. Vasquez: “Chern numbers of cusp resolutions in totally real cubic number fields,” J. Reine Angew. Math. 1981 : 324 (1981), pp. 175–191. MR 614524 Zbl 0491.14018 article

[77]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

Ioan M. James	Related
Alphonse Thomas Vasquez	Related
Peter John Hilton	Related

[78]E. Thomas: “Defects of cusp singularities,” Math. Ann. 264 : 3 (1983), pp. 397–411. MR 714112 Zbl 0504.14002 article

Let \( X \) be a normal complex space and \( x\in X \) a singularity. Following [Knöller 1973, 1982], for each positive integer \( q \) we define defect \( \delta_x(q) \) as follows. Let \( f:\tilde{X}\to X \) be a resolution of \( x \), with \[ S = f^{-1}(x)\subset \tilde{X} .\] Let \( U \) be a small Stein neighborhood of \( x \), \( \tilde{U} = f^{-1}(U) \) and \[ \mathcal{H} = \Omega_{\tilde{X}}^n \] the sheaf of holomorphic forms of degree \( n \) on \( \tilde{X} \). Set \begin{equation*}\tag{1} \delta_x(q) = \dim_C\Gamma(\tilde{U}-S,\mathcal{H}^q)/\Gamma(\tilde{U},\mathcal{H}^q) \quad\in N. \end{equation*} Thus \( \delta_x \) is a measure of the extendability of forms over the exceptional fiber \( S \). Clearly these integers are independent of the choices made in the definition. As in [Knöller 1973] we call \( \delta_x(q) \) the \( q \)-th defect at \( x \). Define the defect series for \( x \), a formal power series in \( \lambda \), by: \[ \Delta(x,\lambda) = 1 + \sum_{q=1}^{\infty}\delta_x(q)\lambda^q \quad\textrm{ in } Z[\![\lambda]\!]. \] We compute the series for an infinite family of cusp singularities arising from totally real cubic number fields and show that it is a rational function.

[79]E. Thomas and A. T. Vasquez: “Rings of Hilbert modular forms,” Compos. Math. 48 : 2 (1983), pp. 139–165. MR 700001 Zbl 0515.10027 article

[80]E. Thomas: “On the zeta function for function fields over \( \mathbf{F}_p \),” Pac. J. Math. 107 : 1 (1983), pp. 251–256. MR 701822 Zbl 0518.12007 article

[81]E. Thomas and A. T. Vasquez: “A family of elliptic curves and cyclic cubic field extensions,” Math. Proc. Camb. Philos. Soc. 96 : 1 (July 1984), pp. 39–43. MR 743699 Zbl 0555.14010 article

[82]E. Thomas: “Complete solutions to a family of cubic Diophantine equations,” J. Number Theory 34 : 2 (February 1990), pp. 235–250. MR 1042497 Zbl 0697.10011 article

[83]E. Thomas: “Solutions to certain families of Thue equations,” J. Number Theory 43 : 3 (March 1993), pp. 319–369. MR 1212687 Zbl 0774.11013 article

[84]E. Thomas: “Solutions to infinite families of complex cubic Thue equations,” J. Reine Angew. Math. 1993 : 441 (January 1993), pp. 17–32. MR 1228609 Zbl 0780.11014 article

[85]E. Thomas: “Counting solutions to trinomial Thue equations: A different approach,” Trans. Am. Math. Soc. 352 : 8 (2000), pp. 3595–3622. MR 1641119 Zbl 0995.11025 article

We consider the problem of counting solutions to a trinomial Thue equation–that is, an equation \begin{equation*}\tag{1} |F(x,y)| = 1, \end{equation*} where \( F \) is an irreducible form in \( Z[x,y] \) with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri–Mueller and Bombieri–Schmidt, all concerned with the “Thue-Siegel principle” and its relation to (1). In this paper we give specific numerical bounds for the number of solutions to (1) by a somewhat different approach, the difference lying in the initial step–solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.

[86]E. Thomas: “Characteristic classes and differentiable manifolds,” pp. 113–187 in Classi caratteristiche e questioni connesse [Characteristic classes and related questions] (L’Aquila, Italy, 2–10 September 1966). Edited by E. Martinelli. C.I.M.E. Summer Schools 41. Springer (Heidelberg), 2010. MR 2768044 incollection

P. Emery Thomas

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