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Cited by 219Advanced Algebra and GeometryGeometric and Algebraic TopologyHomotopy and Cohomology in Algebraic Topology

The DOI matches its Crossref record; 103 of the 104 references it cites were found in a registry or an index; 10 of the 119 sources its claims cite were checked for support.

Topology of Lie groups

Open access
Samelson, Hans·1952·Bulletin of the American Mathematical Society
DOI 10.1090/s0002-9904-1952-09544-6
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Overview

AI snapshot

Written from the paper’s full text.

TL;DR

This 1952 survey sketches the topology of Lie groups, covering classification, covering groups, Lie algebras, compact and noncompact decompositions, Betti numbers, cohomology rings, Klein spaces, and homotopy. It synthesizes known results rather than presenting a new empirical study.

Key findings

  • Connected compact simple Lie groups are classified as the classical orthogonal, unitary, and symplectic families plus five exceptional groups of dimensions 14, 52, 78, 133, and 248.
  • For compact semisimple groups, Weyl’s theorem gives a finite fundamental group and zero first Betti number; Hopf’s theorem describes the cohomology ring as a Grassmann algebra on odd-dimensional primitive generators, with the number of generators equal to the rank.
  • Noncompact connected groups decompose topologically as a product of a maximal compact subgroup and a Euclidean-like factor; solvable groups decompose as a torus times Euclidean space.
  • Torsion results include: unitary and symplectic groups have no torsion, while orthogonal groups have only order-two torsion; the mod 2 Poincaré polynomial of \(O^+(n)\) is given as \((1+t)(1+t^2)\cdots(1+t^{n-1})\). Only \(S^1\) and \(S^3\) among spheres are Lie groups.

Methods

  • Expository survey/address, not an empirical paper; it reviews developments in the topology of Lie groups after Cartan’s earlier account.
  • Organized around definitions, covering groups, Lie algebras, classification, compact and noncompact groups, Betti numbers, cohomology theories, Klein spaces, and homotopy properties.
  • No data or sample; the main mode is synthesis of cited literature, with proof sketches for selected results.

Limitations

  • Stated: the author is only incidentally concerned with general topological groups and explicitly excludes Hilbert’s fifth problem; overlap with Cartan’s account is unavoidable.
  • Stated: in the algebraic-methods section, definitions and proofs cannot be given; some problems remain incomplete, such as the case of a subgroup homologous to zero in \(G\), and higher homotopy groups are far from complete.
  • Evident from the text: as a survey, many results are cited without full proof, and the work does not present new empirical data or a systematic primary analysis.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1090/s0002-9904-1952-09544-6

Cites

103 of 103 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Verification

Claims the paper makes

10 of 119 cited sources checked · 1 supported · 0 flagged · accuracy not yet measured

  • Premisecitation

    A local homomorphism from a simply connected Lie group can always be extended to a homomorphism of the whole group.

    “A local homomorphism of a simply connected group can always be extended to a homomorphism of the whole group [107; 108; 21; 96].”
    Quote in the paperNot a substantial claim1 of 4 sources checked
    • cites [107]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [108]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [21] 21. S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. vol. 45 (1944) pp. 747-752. source
    • cites [96] 96. G. D. Mostow, A new proof of Ê. Cartan y s theorem on the topology of semi-simple groups, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 969-980. source

      The proof of Lemma 2.7 discusses a simply connected homogeneous space and a topological decomposition of a semisimple Lie group, not extension of local homomorphisms. Searches for “local homomorphism” and for extension to a global homomorphism found no passage in our copy addressing the claim. (LLM, conf=0.74)

      Experimental — accuracy not yet measured
  • Premisecitation

    For H a circle group S^1, the Poincaré polynomial of G/H is obtained from that of G by replacing a factor 1+t^3 by 1+t^2.

    “it states that the Poincaré polynomial of G/H is obtained from that of G by replacing a factor 1+/ 3 by a factor 1+/ 2 .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [56] 56. Y. Kawada, On the invariant differential forms of local Lie groups, J. Math. Soc. Jap. vol. 1 (1949) pp. 217-225. source

      Theorem 2 (Additional information; page unavailable) gives a factorization of the Poincaré polynomial of G, but does not address the polynomial of G/H. Searches for a circle subgroup, G/H, and the proposed replacement of 1+t³ by 1+t² found no passage stating that relationship. (LLM, conf=0.74)

      Experimental — accuracy not yet measured
  • Interpretationcitation

    Any compact group can be obtained as a product of finitely many compact, simply connected, simple, semisimple groups and a torus group, modulo a finite normal subgroup.

    “Using Weyl's theorem it follows then easily that any compact group can actually be obtained as product of (a finite number of) compact, simply connected, simple, semi-simple groups and a torus group, modulo a finite normal subgroup [13; 96].”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [13] 13. R. Brauer, Sur les invariants intégraux des variétés des groupes de Lie simple clos, C. R. Acad. Sci. Paris vol. 201 (1935) pp. 419-421. source
    • cites [96] 96. G. D. Mostow, A new proof of Ê. Cartan y s theorem on the topology of semi-simple groups, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 969-980. source

      In the proof of Theorem 2 (page not available), Mostow decomposes a connected group’s maximal compact subgroup using a compact semisimple part and a toroidal part. Our copy does not give the claimed decomposition into finitely many simply connected simple groups modulo a finite normal subgroup, and its result does not extend to any compact group. (LLM, conf=0.84)

      Experimental — accuracy not yet measured
  • Premisecitation

    Most known results on higher homotopy groups of Lie groups come from groups acting transitively on spheres, via fibre space relations.

    “the reason for this is that most of the existing results are obtained for groups acting transitively on spheres, by fibre space relations [29; 30; 126; 133; 134; 136; 116].”
    Quote in the paperNot a substantial claim1 of 7 sources checked
    • cites [29] 29. C. Chevalley, Determination of the Betti numbers of the exceptional Lie groups, Proceedings of the International Congress of Mathematicians, 1950 (to appear).

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [30] 30. D. van Dantzig, and B. L. van der Waerden, Über metrisch homogene Raume, Abh. Math. Sem. Hamburgischen Univ. vol. 6 (1928) pp. 367-376. source
    • cites [126] source

      The passage defining the projection from the rotation group to the sphere (section labelled “Abstract”; page unavailable) gives a sphere as a coset space and describes the associated fibering. The paper also gives a result concerning a sphere bundle and a homotopy group. It does not say that most existing results on higher homotopy groups of Lie groups were obtained this way. (LLM, conf=0.81)

      Experimental — accuracy not yet measured
    • cites [133] source
    • cites [134]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [136]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [116]
  • Premisecitation

    For spheres of even dimension, the isotropy group of a transitive transformation group must be a maximal subgroup of maximal rank.

    “For spheres of even dimension another approach has been given by A. Borel [7]: the sphere has characteristic two, and two is a prime number; this implies easily that the isotropy group must be a maximal subgroup of maximal rank, and now one can make use of the complete list of such groups in §25.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [7] 7. A. Borel, Some remarks about transformation groups transitive on spheres and tori, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 580-586. source
  • It may state two separate claims as one. 44
  • Probably not a substantial claim. 10

  • It may state two separate claims as one.

    Premisecitation

    Weyl's theorem says that the fundamental group of a compact semisimple group is finite, equivalently its first Betti number is zero, and its universal covering group is compact.

    “The last statement is given more precision by an important theorem of H. Weyl, saying that the fundamental group of a compact semi-simple group is finite (or that the first Betti number is zero), in other words, that the universal covering group is still compact.”
    Quote in the paperSubstantial claim1 of 5 sources checked
    • cites [129] source
    • cites [13] 13. R. Brauer, Sur les invariants intégraux des variétés des groupes de Lie simple clos, C. R. Acad. Sci. Paris vol. 201 (1935) pp. 419-421. source
    • cites [25] 25. On the topological structure of solvable groups, Ann. of Math. vol. 42 (1941) pp. 668-675.
    • cites [43] 43. M. Goto, Faithful representations of Lie groups I, Mathematica Japonicae vol. 1 (1949) pp. 107-119. source

      In the proof following Theorem 3 (page unavailable), Gotô invokes Weyl’s theorem to say that the compact form of a complex semisimple Lie algebra generates a compact group, then discusses a simply connected group generated by that form. The copy does not state the claimed finiteness of the fundamental group or the equivalence with a zero first Betti number, and it does not explicitly say that the universal covering group of every compact semisimple group is compact. (LLM, conf=0.75)

      Experimental — accuracy not yet measured
    • cites [105] source

  • It may state two separate claims as one.

    Premisecitation

    Pontryagin showed that the unitary and symplectic groups have no torsion, and all torsion elements of the orthogonal group have order two.

    “Pontryagin has shown in his explicit construction [99] that the unitary and symplectic group have no torsion at all, and that all the torsion elements of the orthogonal group are of order two.”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [99] 99. J. Nordon, Les éléments d'homologie des quadriques et des hyperquadriques, Bull. Soc. Math. France vol. 74 (1946) pp. 11-129. source

      Section 3, “Groupes d’homologie,” discusses torsion in the homology of quadrics, not torsion in unitary, symplectic, or orthogonal groups. Searches for Pontryagin and for those groups’ torsion found no passage in our copy stating the attributed result. (LLM, conf=0.76)

      Experimental — accuracy not yet measured

  • It may state two separate claims as one.

    Premisecitation

    If a simply connected Klein space has characteristic 1, it is a single point; if characteristic 2, it is a sphere; if characteristic is a prime greater than 2, it is one of a small list of spaces.

    “If a (simply connected) Klein space has characeristic 1, it is a single point [127]; if it has characteristic 2, it is a sphere; if it has characteristic a prime>2, it is one of a small list of spaces [7; 127].”
    Quote in the paperNot a substantial claim1 of 2 sources checked
    • cites [127] source
    • cites [7] 7. A. Borel, Some remarks about transformation groups transitive on spheres and tori, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 580-586. source

      Theorems IV and V in §2 support the sphere and finite-list claims for simply connected spaces on which a compact connected Lie group acts transitively. Theorem IV specifies even-dimensional spheres. Our copy does not state that characteristic 1 implies a single point, and the citing sentence omits the restriction on the acting group. (LLM, conf=0.86)

      Experimental — accuracy not yet measured

  • It may state two separate claims as one.

    Premisecitation

    Except for three cases, only the well-known groups can act transitively on a sphere: for even dimensions only the orthogonal group plus G2, for dimension 4n+1 the orthogonal, unimodular unitary, and unitary groups, and for dimension 4n-1 also the symplectic and full quaternion-unitary groups.

    “Except for three cases only those groups can act transitively on a sphere which are well known to do so, namely for even dimensions only the orthogonal group (plus G 2 , which has 5 6 as Klein space), for dimension 4w+l the orthogonal, the unimodular unitary, and the unitary group, for dimension 4n-1 also the symplectic group and the full quaternion-unitary group (plus 5 16 = B4/B3 and 5 7 = S 3 /”
    Quote in the paperSubstantial claim1 of 3 sources checked
    • cites [7] 7. A. Borel, Some remarks about transformation groups transitive on spheres and tori, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 580-586. source

      Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

      Experimental — accuracy not yet measured
    • cites [8] 8. Le plan projectif des octaves et les sphères comme espaces homogènes, C. R. Acad. Sci. Paris vol. 230 (1950) pp. 1378-1380.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [89] 89. On the discrete subgroups and homogeneous spaces of nilpotent Lie groups, Nagoya Math. J. vol. 2 (1951) pp. 95-110.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisenone given

    Ideals in a Lie algebra correspond to normal subgroups, and Lie algebra homomorphisms correspond to local homomorphisms of the associated groups.

    “ideals correspond to normal subgroups; homomorphism of the Lie algebra and local homomorphism of the group correspond to each other.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    The Lorentz group is homeomorphic to the product of its orthogonal subgroup and the set of positive definite symmetric matrices, and that latter set is homeomorphic to Euclidean 8-space.

    “That means that the Lorentz group is homeomorphic to the Cartesian product of its subgroup of orthogonal matrices and the subset of positive definite symmetric matrices; and the latter set is easily shown to be homeomorphic with Euclidean £ 8 .”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisenone given

    Two Lie groups whose Lie algebras are isomorphic are locally isomorphic, so almost all properties of a Lie group are contained in its Lie algebra.

    “The principal fact concerning the Lie algebra is that two groups with isomorphic Lie algebras are locally isomorphic, and that therefore almost all properties of a group must be contained in the Lie algebra.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisecitation

    All simple Lie algebras and simple Lie groups are known.

    “A fact of very great importance is that all simple Lie algebras and groups are known (the construction of a general Lie group is reduced thereby to an-in general nontrivial-extension problem).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation

    The Levi-Whitehead theorem says that any Lie algebra has a semisimple subalgebra mapping isomorphically onto the factor algebra modulo the radical, giving a split extension.

    “The important Levi-Whitehead theorem [135] says that any Lie algebra has a semi-simple subalgebra which maps isomorphically on the factor algebra modulo the radical, so that one has a split extension.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [135] source

  • It may state two separate claims as one.

    Interpretationreasoning

    Any compact group is locally isomorphic to a direct product of finitely many simple groups; more precisely, it is obtained from the direct product of their simply connected covering groups by dividing by a discrete normal subgroup.

    “If one pursues the above reasoning somewhat further, one arrives at the fact that any compact group is locally isomorphic with the direct product of a finite number of simple groups, and more precisely is obtained from the direct product of the simply connected covering groups of these simple groups by dividing by a discrete normal subgroup.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    All extensions of one compact group by another are locally isomorphic and share the same universal covering group; they differ by discrete normal subgroups.

    “This means that all extensions of one compact group by another are locally isomorphic, and so they all have the same universal covering group; they are distinguished by the different discrete normal subgroups by which one has to divide.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisecitation

    Mostow also showed, for simply connected G, that the subgroup generated by a subalgebra M is closed if dim L - dim M < 5; for larger differences this is not always so.

    “He also showed, under the same hypothesis on G, that the subgroup, generated by the subalgebra M y is closed if dim L -dim M<5; for larger differences this is not always so.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [93] 93. D. Montgomery and H. Samelson, Transformation groups of spheres, Ann. of Math. vol. 44 (1943) pp. 454-470. source

  • It may state two separate claims as one.

    Premisecitation

    Any connected Lie group is homeomorphic to the Cartesian product of a maximal compact subgroup K and a subset H homeomorphic to Euclidean space; G can be written uniquely as KH, and all maximal compact subgroups are conjugate.

    “Any connected group is homeomorphic to the Cartesian product of a maximal compact subgroup K (which is connected) and a subset H, which is homeomorphic with a Euclidean space; the theorem says more, namely, that there exist 1-parameter subgroups hi, • • • , h r , isomorphic to the real line R, such that H can be written”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [92] 92. A theorem on Lie groups, Bull. Amer. Math. Soc. vol. 48 (1942) pp. 448-452.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    Every Lie group has a unique simply connected universal covering group, and the kernel of the covering projection is discrete and isomorphic to the fundamental group.

    “Every group G possesses a unique simply connected (i.e.all closed paths are contractible to a point) “universal” covering group G; the-discrete-kernel of the projection is isomorphic with the fundamental group (cf.§6) of G.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [108]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [116]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisereasoning

    If H is a closed subgroup of a Lie group G, then G/H is an analytic manifold of dimension dim G - dim H, and the natural projection G to G/H is analytic.

    “the Lie property of G and H imply that G/H is an analytic manifold, whose dimension is the difference of those of G and H; the natural map p of G into G/H, which sends g into the coset gH, the projection, is analytic.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisenone given

    Conjugation of T^l by elements of its normalizer determines a finite group Φ of automorphisms of T^l, and this group as a transformation group of the torus determines the Lie algebra of G.

    “Conjugation of T l with elements of NT determines then a finite group $ of automorphisms of T l .”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    For semisimple groups, compactness is a property of the Lie algebra; this is not true for other types of groups, as S1 and R show.

    “A noteworthy fact about Weyl's theorem is that compactness for semi-simple groups appears as a property of the Lie algebra; this is of course not so for other types of groups, as the groups S 1 and R show.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisecitation

    The fundamental group of the rotation group of 3-space has order two; this holds for all orthogonal groups O+(n), n>2.

    “It is well known here that the fundamental group of the rotation group of 3-space is of order two; actually this is so for all orthogonal groups 0 + (n), n>2 [116].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [116]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    If K is connected, then π1(G/K) is an abelian factor group of π1(G); in general it has π1(G/K0) as an abelian normal subgroup with index equal to the cardinality of K/K0.

    “If K is connected, then TTI(G/K) is easily seen to be a factor group of TTI(G) and therefore abelian ; in the general case it has iri(G/Ko) as abelian normal subgroup, with index equal to the cardinality of K/K 0 (cf.[109]).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [109]

  • It may state two separate claims as one.

    Premisecitation

    For a compact semi-simple group, the second Betti number is always zero and the third is always positive.

    “for a compact semi-simple group the second Betti number is always zero and that the third is always positive [13 (IS)]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [13 (IS)]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisenone given

    The Poincaré polynomials of the classical groups are given by the products P_An=(1+t^3)(1+t^5)...(1+t^{2n+1}), P_Bn=(1+t^3)(1+t^7)...(1+t^{4n-1}), P_Cn=(1+t^3)(1+t^7)...(1+t^{4n-1}), and P_Dn=(1+t^3)(1+t^7)...(1+t^{4n-5})(1+t^{2n-1}).

    “P An = (1 + * 3 )(1 + P) • • • (1 + / 2 «+i), P Bn = (1 + / 3 )(1 + t?) • • • (1 + *«-i) f Pc n = (1 + / 3 )(1 + * 7 )“-(1 + * 4 *-1 ), PD H = (1 + * 3 )(1 + * 7 ) • • • (1 + * 4 “-fi )-(l + ***-*)-”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Premisecitation

    When G/H is an even-dimensional sphere S^m, one primitive generator of the ring of H has dimension m-1, and the ring of G is obtained by replacing this generator by one of dimension 2m-1.

    “when G/H is a sphere S m of even dimension ; the result is that one of the primitive generators of the ring of H has dimension m -1, and that the ring of G is obtained by replacing this generator by one of dimension 2m -1.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [104] 104. Mat. Sbornik N.S. vol. 1 (1936) p. 779. 130. H. Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I, II, III, Math. Zeit. vol. 23 (1925) pp. 271-309 and vol. 24 (1926) pp. 328-395. 131. , The classical groups, Princeton, 1946. 132. , On the structure and representations of continuous groups I, II, mimeo- graphed notes, Institute for Advanced Study, Princeton, 1933-1934, 1934-1935. 133. G. W. Whitehead, Homotopy properties of the real orthogonal groups, Ann. of Math. vol. 43 (1942) pp. 132-146. 134. , On the homotopy groups of spheres and

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    The natural restriction map from invariant symmetric forms of G to those of H completely determines the cohomology ring of G/H, which is constructed by a boundary operator in IA(G)⊗Is(H).

    “This mapping completely determines the cohomology ring of G/H; the ring is constructed by suitable definition of a boundary operator in the tensor product IA{G) ®IS(H).”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [16] 16. É. Cartan, La topologie des groupes de Lie, Acutalités Scientifiques et In- dustrielles, no. 358, Paris, 1936, also L'Enseignement Mathématique vol. 35 (1936) pp. 177-200.
    • cites [17] 17. Selecta, Jubilé Scientifique, Paris, Gauthier-Villars, 1939.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [60] 60. Sur Vhomologie et la cohomologie des algèbres de Lie, C.R. Acad. Sci. Paris vol. 228 (1949) pp 288-290.
    • cites [71] 71. Application continue commutant avec les éléments d'un groupe de Lie compact, C.R. Acad. Sci. Paris vol. 228 (1949) pp. 1784-1786.

  • It may state two separate claims as one.

    Premisenone given

    If G is simple, the polyhedron is a simplex, and the classification of simple groups reduces to simplices satisfying the arithmetical condition on the angles.

    “If the group is simple, the polyhedron is a simplex, and the determination of the simple groups is reduced to the discussion of simplices satisfying the arithmetical condition on the angles.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationcitation

    The Lefschetz fixed point formula implies that the characteristic equals the number of fixed points and is therefore non-negative.

    “one reads off from the Lefschetz fixed point formula (the nontrivial part) [2 ] that the characteristic equals the number of fixed points and is therefore non-negative.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [2] 2. P. S. Alexandroff and H. Hopf, Topologie I, Berlin, 1935. source

  • It may state two separate claims as one.

    Premisenone given

    The fundamental domains of T are congruent convex polyhedra that cover E^l without overlapping, and the angles between two faces are of the form π/k with integer k at least 2.

    “The fundamental domains of T are convex polyhedra, which are all congruent to each other, and cover E l without overlapping; the angles between two faces of a polyhedron are of the type r/k, k integral and not less than 2.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisecitation

    The homology group with Pontryagin multiplication is also a Grassmann algebra, dual to the cohomology Grassmann algebra.

    “One can show in addition that the homology group, with Pontryagin multiplication, is also a Grassmann algebra, and that these two Grassmann algebras are duals of each other [59; 62; 104],”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [59] 59. Sur Vhomologie des espaces homogènes, C.R. Acad. Sci. Paris vol. 225 (1947) pp. 477-479. [January
    • cites [62] 62. Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France vol. 78 (1950) pp. 65-127.
    • cites [104] 104. Mat. Sbornik N.S. vol. 1 (1936) p. 779. 130. H. Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I, II, III, Math. Zeit. vol. 23 (1925) pp. 271-309 and vol. 24 (1926) pp. 328-395. 131. , The classical groups, Princeton, 1946. 132. , On the structure and representations of continuous groups I, II, mimeo- graphed notes, Institute for Advanced Study, Princeton, 1933-1934, 1934-1935. 133. G. W. Whitehead, Homotopy properties of the real orthogonal groups, Ann. of Math. vol. 43 (1942) pp. 132-146. 134. , On the homotopy groups of spheres and

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    Leray determined for the classical groups the ring of G/T, and more generally the ring of G/H when H has the same rank as G and contains T.

    “He has determined, for the classical groups, the ring of G/T, where T is a maximal toral subgroup of G, and more generally the ring of G/H, when H has the same rank as G, i.e., when it contains T.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [69] 69. Sur l'anneau d'homologie de l'espace homogène, quotient d'un groupe clos par un sous-groupe abélien connexe maximum, C.R. Acad. Sci. Paris vol. 223 (1946) pp. 412-415.

  • It may state two separate claims as one.

    Premisecitation

    If H does not bound in G, then G behaves homologically over the rationals as the Cartesian product of H and G/H; the cohomology ring of G/H is a Grassmann algebra with odd-dimensional generators, and the ring of G is the tensor product of the rings of G/H and H.

    “G behaves homologically (over the rationals) as if it were the Cartesian product of H and G/H”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [59] 59. Sur Vhomologie des espaces homogènes, C.R. Acad. Sci. Paris vol. 225 (1947) pp. 477-479. [January
    • cites [104] 104. Mat. Sbornik N.S. vol. 1 (1936) p. 779. 130. H. Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I, II, III, Math. Zeit. vol. 23 (1925) pp. 271-309 and vol. 24 (1926) pp. 328-395. 131. , The classical groups, Princeton, 1946. 132. , On the structure and representations of continuous groups I, II, mimeo- graphed notes, Institute for Advanced Study, Princeton, 1933-1934, 1934-1935. 133. G. W. Whitehead, Homotopy properties of the real orthogonal groups, Ann. of Math. vol. 43 (1942) pp. 132-146. 134. , On the homotopy groups of spheres and

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    Brauer constructed explicit invariants for each classical group that exhaust all possible invariants, matching Pontryagin's results.

    “By use of the first main theorem of invariant theory, Brauer was able to construct, for each classical group, explicitly a certain number of invariants, which from other considerations could be shown to exhaust all possible invariants (cf.also [130; 131 ]).The result corresponded completely to Pontryagin's,”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [130]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [131] source

  • It may state two separate claims as one.

    Interpretationreasoning

    In one proof of that result, the isotropy group must collapse to the identity element, or at least be finite.

    “For one of the proofs one shows that because of the size of the Betti numbers of T n , i.e., because of the large number of invariant differential forms, the isotropy group has to reduce to the unit element (or at any rate be finite)”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisecitation

    Chern determined the mod 2 cohomology ring structure of M_{n,k}, which is a universal antecedent of the characteristic ring of any sphere bundle.

    “Chern [19] has determined the cohomology ring structure of M n , k (with coefficients mod 2) ; the interest in this ring is due to the fact that it is a universal antecedent of the so-called characteristic ring of any sphere bundle”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [19] 19. Notions d'algèbre différentielle] application aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de topologie (espaces fibres), Bruxelles, 1950; Liège and Paris, 1951.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    For Grassmann manifolds, the cells are defined by Schubert varieties, and all such cells form a generalized cell decomposition of M_{n,k}.

    “For the Grassmann manifolds, e.g., the cells are defined by the so-called Schubert varieties [34; 35]”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [34] 34. Système von Richtungsfeldem in Sphdren, Comment. Math. Helv. vol. 15 (1942-1943) pp. 1-26.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [35] 35. C. Ehresmann, Les invariants intégraux et la topologie de Vespace projectif réglé, C.R. Acad. Sci. Paris vol. 194 (1932) pp. 2004-2006.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    π_{m-k}(V_{m,k}) is infinite cyclic when m-k is even or k=1, and of order two when m-k is odd and k>1.

    “they are infinite cyclic if m -k is even or k = 1, of order two if m -k is odd and *>1.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [116]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationreasoning

    When N is normal, π2(G/N) is zero and therefore π1(N) is zero.

    “if N is normal, then v^G/N) is zero, and so is then wi(N).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

    Premisenone given

    The fundamental group of a Lie group is always abelian, and it is finite when the group is compact semi-simple.

    “That the fundamental group of a group is always abelian and that it is finite for compact semi-simple groups was mentioned earlier.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Premisecitation

    Kudo's theorem says that under the stated assumption on W, the isotropy group H is not homologous to zero in G; this is also contained in Koszul's results.

    “An important role is played by a theorem of Kudo which says that under the stated assumption on W the isotropy group H is not ~0 in G ; this is also contained in Koszul's results ( §22).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [84] 84. On the theory of the Lie groups in the large t Mat. Sbornik N.S. vol. 16 (1945) pp. 163-189.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisenone given

    The quotient O(m)/O(m-k) is the Stiefel manifold V_{m,k} of k-frames in Euclidean m-space.

    “The manifold 0(m)/0(m -k)Xlk f which appeared here, can also be described as the space, each point of which is a system of k mutually orthogonal unit vectors at the origin in Euclidean m-space (a &-frame) ; it is called the Stiefel manifold V m ,h.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.

    Premisenone given

    Every compact Lie group has a faithful representation and so embeds as a subgroup of some orthogonal group.

    “G can be considered as subgroup of some orthogonal group 0(k) (since it has a faithful representation).”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Premisecitation

    Iwamoto determined the restriction to orthogonal groups and obtained a rule for trivial representations or linear invariants that expresses the pth Betti number in terms of partitions of p.

    “Iwamoto [51 ], using the known theory of representations of the orthogonal groups, determined what happens when one now restricts oneself to the orthogonal groups.One arrives at a fairly simple rule for the appearance of trivial representations or linear invariants, expressing the pth Betti number in terms of partitions of p.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [51] 51. Maximale Toroide und singuldre Elemente in geschlossenen Lie'schen Gruppen, Comment. Math. Helv. vol. 15 (1942-1943) pp. 59-70.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    The isometry group of a compact connected two-point homogeneous metric space has no arbitrarily small normal subgroups, so it is a compact Lie group and the space is a Klein space of finite dimension.

    “Wang showed by a simple ingenious argument that the group of isometries cannot have arbitrarily small normal subgroups, and therefore must be a (compact) Lie group, and the space is therefore a Klein space (of finite dimension, say n).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [128] source

  • It may state two separate claims as one.

    Premisecitation

    The two-point homogeneous spaces are completely classified: for odd n only spheres and projective spaces occur; for even n complex projective spaces, quaternion projective spaces, and the 16-dimensional Cayley projective plane occur as well.

    “One obtains a complete determination of all possibilities; the spaces occurring are: for odd n only the spheres and projective spaces, for even n there are in addition complex projective spaces, quaternion projective spaces, and the (16-dimensional) projective plane over the Cayley numbers.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [128] source

  • It may state two separate claims as one.

    Premisecitation

    Up to a small finite number of undecided cases, any such space is either a sphere or the space of unit tangent vectors to an even-dimensional sphere, and the groups act in standard fashion.

    “up to a small, finite number of cases, which remain undecided, any such space is either actually a sphere or-and this is the only new case-the space of unit tangent vectors to a sphere of even dimension”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [84] 84. On the theory of the Lie groups in the large t Mat. Sbornik N.S. vol. 16 (1945) pp. 163-189.

  • Probably not a substantial claim. It may state two separate claims as one.

    Premisenone given

    The Weil algebra W(L) is the tensor product of the Grassmann algebra of skew forms on L and the symmetric algebra of symmetric forms on L, and it is the algebraic equivalent of a classifying space for principal fibre bundles with fibre G.

    “it is the tensor product of the Grassmann algebra A (L) of skew forms on L and the algebra S(L) of symmetric forms on L.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation

    The group Γ is generated by reflections in certain planes.

    “The group T is generated by reflections in certain (infinitely many of course) planes; the planes form the so-called diagram of the group [l2l]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [121] source

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation

    Bi-invariant forms coincide with the harmonic forms for the invariant metric.

    “the bi-invariant forms turn out to be identical with the harmonic forms, relative to the invariant metric of the group, and can be treated from this point of view [SO]”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    If a cross section existed, O(n+1) would be the Cartesian product of O(n) and S^n, and S^n would admit a parallelism.

    “If a cross section existed, then, as one sees easily, 0(n+l) would be the Cartesian product of 0(n) and 5 n , and, more interesting, there would exist n everywhere independent vector fields, tangent to 5 W , i.e. in the language of topology, S n would admit a parallelism.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

    Premisecitation

    For a compact Lie group all Stiefel-Whitney characteristic classes vanish and the Euler-Poincaré characteristic is zero.

    “[116] that all the Stief el-Whitney characteristic classes vanish, and in particular that the Euler-Poincaré characteristic x(G) (the alternating sum of the Betti numbers, which equals the alternating sum of the numbers of cells of the various dimensions if G is divided up into a complex (cf.[2])) is 0.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [116]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation

    The p-th Betti number equals the number of linearly independent closed p-forms modulo exact forms.

    “By the famous theorem of de Rham [102] (which was inspired by just these questions) the pth Betti number is equal to the number of linearly independent ^-dimensional closed exterior differential forms defined over G modulo those forms which are derivatives of (p -1) -forms.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [102] 102. Sur les nombres de Betti des groupes de Lie, C.R. Acad. Sci. Paris vol. 200 (1935) pp. 1277-1280.

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Premisenone given

    Spheres of even dimension admit no nonzero vector field.

    “It is an old story that spheres of even dimension do not admit a single (nonzero) vector field.”
    Quote in the paperNot a substantial claim
50 of 50

  • The authors state this as an assumption.

    In the converse problem of which groups can act transitively on a Klein space W, the acting group is assumed to be effective and compact.

    “(We assume the group to be effective, i.e., only the unit e gives the identity transformation of W, and compact.)”
1 of 1

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • The cross-section question names the wrong projection target

    Minor

    The earlier description makes the sphere the coset space of O(n+1) by O(n). The cross-section condition must therefore use the projection onto the sphere, not onto O(n); as written, its composition cannot be the identity on the sphere.

1 of 1

Citations

Cites

Flagged first, then by citations

1 of the 104 references couldn’t be matched to a known paper.

Research works printed but not linked: 51

  1. Borel, Armand, 1948, Sur les sous-groupes fermés connexes de rang maximum des groupes de Lie clos, C. R. Acad. Sci. Paris, 226, 1662
  2. Borel, Armand, 1950, Le plan projectif des octaves et les sphères comme espaces homogènes, C. R. Acad. Sci. Paris, 230, 1378
  3. Borel, Armand, 1950, Impossibilité de fibrer une sphère par un produit de sphères, C. R. Acad. Sci. Paris, 231, 943
  4. Borel, Armand, 1951, Sur la cohomologie des variétés de Stiefel et de certains groupes de Lie, C. R. Acad. Sci. Paris, 232, 1628
  5. Borel, Armand, 1951, La transgression dans les espaces fibrés principaux, C. R. Acad. Sci. Paris, 232, 2392
  6. Borel, Armand, 1951, Sur la cohomologie des espaces homogènes des groupes de Lie compacts, C. R. Acad. Sci. Paris, 233, 569
  7. Borel, Armand, 1951, Détermination des 𝑝-puissances réduites de Steenrod dans la cohomologie des groupes classiques. Applications, C. R. Acad. Sci. Paris, 233, 680
  8. 10. R. Brauer, Sur les invariants intégraux des variétés des groupes de Lie simple clos, C. R. Acad. Sci. Paris vol. 201 (1935) pp. 419-421.
  9. Calabi, Lorenzo, 1949, Sur les extensions de groupes topologiques, C. R. Acad. Sci. Paris, 229, 413
  10. 13. É. Cartan, La topologie des groupes de Lie, Acutalités Scientifiques et Industrielles, no. 358, Paris, 1936, also L’Enseignement Mathématique vol. 35 (1936) pp. 177-200.
  11. Cartan, Henri, 1948, Sur la cohomologie des espaces où opère un groupe. Notions algébriques préliminaires, C. R. Acad. Sci. Paris, 226, 148
  12. 32. C. Ehresmann, Les invariants intégraux et la topologie de l’espace projectif reglé, C.R. Acad. Sci. Paris vol. 194 (1932) pp. 2004-2006.
  13. 33. C. Ehresmann, Sur la topologie de certaines variétés algébriques, C.R. Acad. Sci. Paris vol. 196 (1933) pp. 152-154.
  14. 35. C. Ehresmann, Sur la topologie de certaines variétés algébriques réelles, J. Math. Pures Appl. vol. 16 (1937) pp. 69-110.
  15. 36. C. Ehresmann, Sur la topologie des groupes simples clos, C.R. Acad. Sci. Paris vol. 208 (1939) pp. 1263-1265.
  16. 37. C. Ehresmann, Sur la variété des génératrices planes d’une quadrique réelle et sur la topologie du groupe orthogonal à n variables, C.R. Acad. Sci. Paris vol. 208 (1939) pp. 321-323.
  17. 38. C. Ehresmann, Sur les espaces localement homogènes, L’Enseignement Mathématique vol. 35 (1936) pp. 317-333.
  18. Gantmacher, Felix, 1939, On the classification of real simple Lie groups, Rec. Math. [Mat. Sbornik] N.S., 5(47), 217
  19. 45. H. Hopf, Sur la topologie des groupes clos de Lie et de leurs généralisations, C.R. Acad. Sci. Paris vol. 208 (1939) pp. 1266-1267.
  20. Koszul, Jean-Louis, 1947, Sur le troisième nombre de Betti des espaces de groupes de Lie compacts, C. R. Acad. Sci. Paris, 224, 251
  21. Koszul, Jean-Louis, 1947, Sur les opérateurs de dérivation dans un anneau, C. R. Acad. Sci. Paris, 225, 217
  22. Koszul, Jean-Louis, 1947, Sur l’homologie des espaces homogènes, C. R. Acad. Sci. Paris, 225, 477
  23. Koszul, Jean-Louis, 1949, Sur l’homologie et la cohomologie des algèbres de Lie, C. R. Acad. Sci. Paris, 228, 288
  24. Koszul, Jean-Louis, 1949, Sur la cohomologie relative des algèbres de Lie, C. R. Acad. Sci. Paris, 228, 457
  25. Leray, Jean, 1946, L’anneau d’homologie d’une représentation, C. R. Acad. Sci. Paris, 222, 1366
  26. Leray, Jean, 1946, Structure de l’anneau d’homologie d’une représentation, C. R. Acad. Sci. Paris, 222, 1419
  27. Leray, Jean, 1946, Propriétés de l’anneau d’homologie de la projection d’un espace fibré sur sa base, C. R. Acad. Sci. Paris, 223, 395
  28. Leray, Jean, 1946, Sur l’anneau d’homologie de l’espace homogène, quotient d’un groupe clos par un sousgroupe abélien, connexe, maximum, C. R. Acad. Sci. Paris, 223, 412
  29. Leray, Jean, 1949, Espace où opère un groupe de Lie compact et connexe, C. R. Acad. Sci. Paris, 228, 1545
  30. Leray, Jean, 1949, Application continue commutant avec les éléments d’un groupe de Lie compact, C. R. Acad. Sci. Paris, 228, 1784
  31. Leray, Jean, 1949, Détermination, dans les cas non exceptionnels, de l’anneau de cohomologie de l’espace homogène quotient d’un groupe de Lie compact par un sous-groupe de même rang, C. R. Acad. Sci. Paris, 228, 1902
  32. Leray, Jean, 1949, Sur l’anneau de cohomologie des espaces homogènes, C. R. Acad. Sci. Paris, 229, 281
  33. Malcev, A., 1942, On the simple connectedness of invariant subgroups of Lie groups, C. R. (Doklady) Acad. Sci. URSS (N.S.), 34, 10
  34. Malcev, A., 1942, Subgroups of Lie groups in the large, C. R. (Doklady) Acad. Sci. URSS (N.S.), 36, 5
  35. Malcev, A., 1942, On the structure of Lie groups in the large, C. R. (Doklady) Acad. Sci. URSS (N.S.), 37, 3
  36. Malcev, A., 1943, On linear Lie groups, C. R. (Doklady) Acad. Sci. URSS (N.S.), 40, 87
  37. Malcev, A., 1944, On semi-simple subgroups of Lie groups, Bull. Acad. Sci. URSS. S\'{e}r. Math. [Izvestia Akad. Nauk SSSR], 8, 143
  38. Mal′cev, A. I., 1949, On a class of homogeneous spaces, Izv. Akad. Nauk SSSR Ser. Mat., 13, 9
  39. 97. L. Pontryagin, On Betti numbers of compact Lie’s groups, C.R. (Doklady) Acad. Sci. URSS. vol. 1 (1935) pp. 433-437.
  40. 98. L. Pontryagin, Sur les nombres de Betti des groupes de Lie, C.R. Acad. Sci. Paris vol. 200 (1935) pp. 1277-1280.
  41. Pontryagin, L. S., 1949, Some topological invariants of closed Riemannian manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 13, 125
  42. Racah, Giulio, 1950, Sulla caratterizzazione delle rappresentazioni irriducibili dei gruppi semisemplici di Lie, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 8, 108
  43. 102. G. de Rham, Sur l’analysis situs des variétés à n dimensions, J. Math. Pures Appl. vol. 10 (1931) pp. 115-200.
  44. Samelson, Hans, 1949, Sur les sous-groupes de dimension 3 des groupes de Lie compacts, C. R. Acad. Sci. Paris, 228, 630
  45. de Siebenthal, Jean, 1950, Sur certaines sous-groupes de rang un des groupes de Lie clos, C. R. Acad. Sci. Paris, 230, 910
  46. Wang, Hsien-Chung, 1949, Homogeneous spaces with non-vanishing Euler characteristic, Acad. Sinica Science Record, 2, 215
  47. Wang, Hsien-Chung, 1949, A new characterisation of spheres of even dimension, Nederl. Akad. Wetensch., Proc., 52, 838
  48. 129. A. Weil, Démonstration topologique d'un théorème fondamental de Cartan, C.R. Acad. Sci. Paris vol. 200 (1935) pp. 518-520
  49. Mat. Sbornik N.S. vol. 1 (1936) p. 779.
  50. Yen, Chih-ta, 1949, Sur les polynomes de Poincaré des groupes simples exceptionnels, C. R. Acad. Sci. Paris, 228, 628
  51. Yen, Chih Ta, 1949, Les représentations linéaires de certains groupes et les nombres de Betti des espaces homogènes symétriques, C. R. Acad. Sci. Paris, 228, 1367

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Metadata

Authors
Samelson, H.
Corresponding author
Samelson, Hans
Venue
Bulletin of the American Mathematical Society
Publisher
American Mathematical Society (AMS)
Published
1952
Volume
58
Issue
1
Pages
2-37
ISSN
  • 0002-9904 (ISSN-L)
  • 1088-9485
  • 0273-0979
DOI
10.1090/s0002-9904-1952-09544-6
OpenAlex
W2071315267
Semantic Scholar
a7ddc9713ab15ebbd71d02f6c6eb8155168c806f
Cited by
219
References
104
Citations by year
2026: 1 · 2025: 1 · 2024: 1 · 2023: 1 · 2022: 8 · 2021: 5 · 2020: 10 · 2019: 3 · 2018: 3 · 2017: 2 · 2016: 7 · 2015: 5 · 2014: 4 · 2013: 8 · 2012: 5
FWCI
2.77
Type
journal article
Language
en
Open access
Locations
  • Bulletin of the American Mathematical Society (publishedVersion)
  • Project Euclid (Cornell University) (submittedVersion)
Topics
  • Physical Sciences › Mathematics › Algebra and Number Theory › Advanced Algebra and Geometry
  • Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
  • Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
Keywords
Lie groups, differential topology, smooth manifolds

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

A language model drew the claims from the paper on 2026-10-02, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).

A second model labelled each claim on 2026-10-02 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).

Cited sources are checked with a language model’s help (gpt-6-sol).

Statistics and arithmetic are recomputed, not judged.

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