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An application of the Morse theory to the topology of Lie-groups
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This paper is primarily devoted to a detailed account of the results announced and sketched in [5 |.It will be followed by a joint paper witli H. SAMELSON on the loop-space of symmetric spaces in general.We recall the principal results of [3].Let G be a compact Lie group; T'C G shall denote a torus in G and C( T) the centralizer of T in G.The space of loops on G is denoted by ^2(G).(For detailed definitions, see §9.) THEOREM A. -If G is a connected^ simply connected^ compact Lie groups then the spaces 12 (G) aud G/C( T') have the following properties : a.They are free of torsion; b.Their odd Betti numbers vanish; c.Their Betti numbers can be read off from the diagram of G. '25l> R. BOTT.THEOREM B. -1° The Poincare series of the loop-space, i2(G'), is given by P(^(G),t)=^t^\where A runs over the cells of the fundamental chamber ^.2° If^y is a lattice point ofF in t, and if C(^Y) is the centralize r of the direction JT, then the Poincare series of G/C(A^) is given bywhere now the summation is extended over the cells of 5 7 which contain X in their closure.As every C(7'' / ) is conjugate to some C(^T), ^T^r, part 2° of this theorem describes the homology of all the spaces G/C(T').It also relates their Poincare series to that of^(G).These formulae can of course be expressed entirely in terms of the rootforms of G on t : Let { 0; \rj (i' = i, . .., /n), denote the root-forms on t which take positive values in 5 7 '.Also let H^ denote the Weyl group ofG in t.Finally, ifJTet, define a function ^*, on W by : A* (w) == number of the root-forms j 6^-J£F whose values at X and w'A" differ in sign.Thus, A *(w) == number of planes of the infinitesimal diagram D'\ crossed by the line from X to w'X.Finally let p(^T) denote the order of the subgroup of W which leaves JT fixed.In terms of these notions, theorem B is equivalent to : THEOREM B'. -i° The Poincare series of^l(G) is given by " 2^[8,W] P^(G)^t)=---j t^ dv, I A 1 ŵhere the integral is taken overly \ A | denotes the volume of a fundamental cell o/Z), and [a] denotes the greatest integer less than a; 2° The Poincare series of G/C(^T), X arbitrary in t, is given by P{(G/CW);t}^--.^t^.w€Î n particular (for X in ^) the iq^ Betti number of G/T is equal to the
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TL;DR
Bott applies Morse theory to the Riemannian geometry of compact connected Lie groups to study path and loop spaces, showing these spaces are torsion-free with vanishing odd Betti numbers. The paper derives Poincaré-series formulas in terms of the group’s diagram and Weyl group, and uses them to compute Betti numbers for exceptional groups.
Key findings
- The path space Ω(G,N,P) is torsion-free, has no odd Betti numbers, and its Poincaré series equals the Morse series of (G,N,P) (Theorem I).
- The Morse series is given by summing t^{λ(s)} over straight-line segments in S(t,N,P), where λ(s) counts how many planes of the diagram D(G) the segment crosses (Theorem II).
- For G/T, the q-th Betti number equals the number of Weyl-group elements that change the sign of exactly q positive root forms (Theorem B′).
- Appendix II computes Betti numbers for the exceptional groups E6, E7, and E8; the text reports E6 dimension 78 and E7 dimension 133, and gives Poincaré polynomials for these groups.
Methods
- The study uses a compact connected Lie group G with a bi-invariant Riemannian metric, and analyzes spaces of paths starting on an adjoint orbit N and ending at a regular point P.
- The adjoint action is shown to be variationally complete, so geodesics reduce to straight lines in a maximal torus t; the Morse index of a segment is computed by counting intersections with the diagram D(G) or D′(G).
- Morse inequalities are used; because the relevant indices are even, the inequalities become equalities, yielding Poincaré series and torsion-freeness results.
Limitations
- The paper states that at the time there was no alternate proof that Ω(G) is free of torsion.
- It notes that the Morse series may fail to exist if its coefficients are infinite; existence is likely when π1(M,N) is finite, but no proof is given and it is not needed for the applications.
- The general variational-completeness case does not give equalities for dimensional reasons; that broader situation is deferred to a subsequent paper with Samelson.
- Evident from the text: the OCR/scan is fragmentary and garbled in places, making some formulas and numerical details difficult to verify.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
0 of 6 cited sources checked · 0 supported · 0 flagged
- Findingreasoning§ 4. THE MORSE SERIES OF M mod TV.
The Morse series has only a countable number of terms.
“In any case the series 4.J has only a countable number of terms.”
Quote in the paperNot a substantial claim - Findingreasoning§ 9. THE DIAGRAM OF G.
If σ in T is on precisely l of the sets Ui, then dim C(σ)=l+2l.
“Further ifo-e T is on precisely/of the sets Ui(i=z i, . .., ni) then dim C (o-) == /+ 2/, and hence (9.2) (W==2/”
Quote in the paperSubstantial claim - Findingreasoning§ A=[^^\^e^}, ^==JA[A€^J.
The function f from A to £ is onto.
“(2)/is onto. Let the closure of A e ^, contain the lattice point Y. Then the closure of y-1 A contains the origin of t, and therefore differs from Ay by an element of W.”
Quote in the paperSubstantial claim - Findingcitation§ 3. FOCAL POINTS.
Let s be a geodesic segment of (M,N). Then the index of s relative to N is always a finite integer.
“PROPOSITION 3.2.-Let s be a geodesic segment of (M^ N).Then the index of 5, relative to TV, defined by (3 9) ^N(s)==^dlmA N (s f ) s'Cs is always a finite integer.”
Quote in the paperSubstantial claim - Findingreasoning§ 9, (JT) == o mod i;. 1=1
If G and G1 are locally isomorphic, such an isomorphism induces corresponding isomorphisms of W onto W1.
“Thus if G and GI are locally isomorphic, such an isomorphism induces corresponding isomorphisms of W^ onto W^_.”
Quote in the paperNot a substantial claim
- Findingreasoning§ 9, (JT) == o mod i;. 1=1
The effect of w in W on t is completely described by the effect of w on one cell of D(G).
“In this case then the effect of w € W on t is completely described by the effect of w on one cell of D(G).”
Quote in the paperSubstantial claim - Findingnone given§ --i /'
The odd Betti numbers of Ω(G,N,P) vanish.
“Its odd Betti numbers vanish, and the Poincare series of Sl(G, N, P) coincides with the Morse series of (G, TV, P).”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
- The authors state this as an assumption: T is a maximal torus of G.
- The authors state this as an assumption: N is an orbit of the adjoint action of G on G.
- The authors state this as an assumption: P is a general point of T not on N.
- Findingnone given§ o'ec'
The indices which occur in the Morse theory here all turn out to be even.
“Secondly, the indices which occur in the Morse theory, here all turn out to be even.”
Quote in the paperSubstantial claim - Findingreasoning§ P^(G)^t)=---j t^ dv, I A 1 ^
π1(G) is a free group.
“An immediate corollary of these theorems is that 7::.,(G) is a free-group.”
Quote in the paperSubstantial claim - Findingnone given§ Theorem B
If γ is a lattice point of Γ in t, and C(γ) is the centralizer of the direction γ, then the Poincare series of G/C(γ) is given by P(G/C(γ);t)=Σ t^{...}, where the sum is over the cells of the fundamental chamber containing γ in their closure.
“If^y is a lattice point ofF in t, and if C(^Y) is the centralize r of the direction JT, then the Poincare series of G/C(A^) is given by P( G/CW ; t) = { ^ t^} t-^W where now the summation is extended over the cells of 5 7 which contain X in their closure.”
Quote in the paperSubstantial claim - Findingreasoning§ 8. THE GEODES1CS OF (G, TV, P).
A∩T is a finite set of points.
“A n T will be a finite set of points because the orbits are compact and meet T at right angles.”
Quote in the paperSubstantial claim - Findingnone given§ 3. FOCAL POINTS.
An element Y in Jg is in J^N if and only if Y0 is in Np and Y0' + Tg Y0 is in Np^⊥.
“PROPOSITION 3.1.-An element Y^Jg is in J^ if and only if (3.6) -To€^, (3.7) ro4-7,.ro€Aa,where : p is the initial point, g{Q)i of gi and Tg is a self-adjoint linear transformation of Np^ determined completely by g.”
Quote in the paperSubstantial claim - Findingnone given§ 3. FOCAL POINTS.
The regular points of (M,N) are everywhere dense in M.
“PROPOSITION 3.3 -The regular points of (M, N) are everywhere dense in M.”
Quote in the paperNot a substantial claim - Findingnone given§ o'ec'
The adjoint action of G on the Lie algebra of G is variationally complete.
“the adjoint action of G, both on G and on the Lie algebra of G, is « variationally complete ».”
Quote in the paperNot a substantial claim - Findingreasoning§ 3. FOCAL POINTS.
dim J^N = dim N.
“An immediate corollary is (3.8) dim7y=dim7V.”
Quote in the paperNot a substantial claim - Findingreasoning§ 9, (JT) == o mod i;. 1=1
On the set {θ_i}, the Weyl group is represented by signed permutations.
“On the set { 9, }y the Weyl group is therefore represented by signed permutations.”
Quote in the paperNot a substantial claim - Findingreasoning§ P^(G)^t)=---j t^ dv, I A 1 ^
If G is simple, then π1(G)=Z.
“If G is simple, then 7T:{(6') == Z.”
Quote in the paperSubstantial claim - Findingnone given§ 9, (JT) == o mod i;. 1=1
The elements of W permute the totality of root forms {±θ_i}.
“The elements of W permute the totality of root forms { ± 9, |.”
Quote in the paperNot a substantial claim - Findingreasoning§ 8. THE GEODES1CS OF (G, TV, P).
If P is a point on an orbit of maximal dimension, then the centralizer of P is precisely a single maximal torus T.
“It follows immediately from (8.4), that if P is a point on an orbit of maximal dimension, then the centralizer of P is precisely a single maximal torus T.”
Quote in the paperSubstantial claim - Findingnone given§ 9, (JT) == o mod i;. 1=1
The orbit of a point P in t under Γ is a lattice of points in t.
“The orbit of a point Pet under F is a lattice of points in t.”
Quote in the paperSubstantial claim - Findingnone given§ o'ec'
The adjoint action of G on G is variationally complete.
“the adjoint action of G, both on G and on the Lie algebra of G, is « variationally complete ».”
- Also stated in: § --i /' · plain
“PROPOSITION 7 . 2 . -The adjoint action of G on G is variationally complete.”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
- Also stated in: § --i /' · plain
- Findingnone given§ 9, (JT) == o mod i;. 1=1
Γ is abstractly isomorphic to π1(T)=Z+Z+...+Z (l factors).
“This latter group, r, is of course abstractly isomorphic to 7:1 ( T) = Z + Z -+-... 4-Z (/ factors).”
Quote in the paperNot a substantial claim - Findingreasoning§ S{Mr\T,Nr\T,P).
For X∈t, G/C(X) is homeomorphic to the orbit of X under the adjoint representation.
“But then G/C(^¥) is homeomorphic to A, the orbit of X under the adjoint representation.”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
- Findingreasoning§ --i /'
Under the conditions of Proposition 12.1, the Poincare series of E∞ dominates the Poincare series of H(Ω) for any coefficient field k.
“COROLLARY 12 . 1 . -Under the conditions of this proposition the Poincare series of Ey dominates the Poincare series of ff(^l) for any coefficient field k.”
Quote in the paperSubstantial claim - Findingnone given§ --i /'
The space Ω(G,N(X),P) is free of torsion for any X∈t.
“THEOREM C. -The space £2(G, N{X)^ P) is free of torsion for any ^€t.”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
- The authors state this as an assumption: T is a maximal torus of G.
- The authors state this as an assumption: N is an orbit of the adjoint action of G on G.
- The authors state this as an assumption: P is a general point of T not on N.
- Findingreasoning§ Additional information
The Poincare polynomial of E6 is (1+t^3)(1+t^9)(1+t^11)(1+t^15)(1+t^17)(1+t^23).
“Hence the 7n/ are given by i, 4, 5, 7, 8, ii, and the Poincare polynomial of E^ is ( I + t 3 ) (I -+-t 9 ) (I -+-{ ii ) ( I + t^) ( I -f-< 17 ) ( I -4-^23 ).”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is simple, connected, simply connected and compact.
- Findingreasoning§ --i /'
For a simple, connected, simply connected compact Lie group G, the relation (13.4) holds among the coefficients of the dominant root, the coefficients of the sum of the positive roots, and the exponents.
“PROPOSITION 13.1.-Let G be simple^ connected^ simply connected and compact.Then the following relation holds l l l (13.4) n[J^J](m,-i)=J]a,.^j 1 Here l is the rank of G, the di are the coefficients of the dominant root of G, the ai are the coefficients of the sum of the positive roots of G and the mi are the exponents of G.”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is simple, connected, simply connected and compact.
- Findingnone given§ --i /'
The exponential map exp|t maps the straight lines of S(t,N,P) in a one-to-one fashion onto the segments of S(G,N,P).
“PROPOSITION 8 . 2 . -The exponential map exp 11, maps the straight lines o»/*S(t, TV, P) in a one-to-one fashion onto the segments of S(G^ .V, P).”
Quote in the paperNot a substantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
- The authors state this as an assumption: T is a maximal torus of G.
- The authors state this as an assumption: N is an orbit of the adjoint action of G on G.
- The authors state this as an assumption: P is a general point of T not on N.
- Findingnone given§ Additional information
The index of s∈S(M∩t,N∩t,P) (properly, of its image in G) is given by (11.1) γ(s) = -2 times the number of planes of D'(G) crossed by s.
“PROPOSITION 11 . 4 . -The index of^€S'(Jifnt, TVnt, P) (properly^ of its image in G) is given by (11.i) y (s) •==-2 times the number of planes of D'(G) crossed by s.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: M is an open spherical disc about the origin in g, so small that exp|M maps M homeomorphically into G.
- Findingnone given§ --i /'
Let G be connected, simply connected and compact. Let F be a fundamental chamber of W on t, and let A^ be the unique cell of D(G) contained in F whose closure contains the origin of t. Then there is a one-to-one correspondence between the cells of the form {r A^ (r∈Γ)} and the cells of the form A∈A.
“LEMMA 9 . 1 . -Let G be connected^ simply connected and compact.Let ^ be a fundamental chamber of W on t, and let A^ be the unique cell of D(G)^ which is contained in ^', and whose closure contains the origin oft.Then there is a one-to-one correspondence between the cells of the form . ^ {r^i (r^r)and the cells of the form b. A€^.”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
- Findingnone given§ Additional information
G/C(X) is free of torsion.
“a. G/C(^¥) is free of torsion;”
Quote in the paperNot a substantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
- Findingnone given§ Additional information
The map exp|M maps the lines of S(M∩t,N∩t,P) onto the segments S(M,N,P) in a one-to-one fashion.
“PROPOSITION 11.3.-The map exp | Jif, maps the lines o/S(J^n t, TVn t, jP), onto the segments S (M^ TV, P) in a one-to-one fashion.”
Quote in the paperSubstantial claim- The authors state this as an assumption: M is an open spherical disc about the origin in g, so small that exp|M maps M homeomorphically into G.
- Findingnone given§ S{Mr\T,Nr\T,P).
The adjoint representation of a compact Lie group on its Lie algebra is variationally complete.
“PROPOSITION 11.6.-The adjoint representation : o-->Ado-of a compact Lie group on its Lie algebra is vocationally complete.”
Quote in the paperNot a substantial claim - Findingreasoning§ 1,. SECOND APPLICATION G/C(A').
The adjoint action of G on M is variationally complete.
“PROPOSITION 11.1. -The adjoint action of G on M is variationally complete.”
Quote in the paperSubstantial claim- The authors state this as an assumption: M is an open spherical disc about the origin in g, so small that exp|M maps M homeomorphically into G.
- Findingreasoning§ Additional information
The relation (13.4), together with the conditions m_i ≥ a_i and m_i=2 for precisely one i, uniquely determines the exponents m_i of E6, E7, and E8.
“The exponents, m^ of a simple group G have the following elementary property {13.13) m/^ a, mi= 2 for precisely one f, relation (13.4), together with these two conditions uniquely determines the exponents mi of E^ E^^ and E^”
Quote in the paperNot a substantial claim- The authors state this as an assumption: G is simple, connected, simply connected and compact.
- Findingreasoning§ Additional information
The Poincare polynomial of E7 is (1+t^3)(1+t^11)(1+t^15)(1+t^19)(1+t^23)(1+t^27)(1+t^35).
“They are therefore the only solutions, and the Poincare polynomial of E^ is given by (i + t 3 ) (i -+-^11 ) (i -t-< 15 ) (i -+-/ l9 ) (i -+-t^) (i + < 27 ) (i -+-< 35 ).”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is simple, connected, simply connected and compact.
- Premisenone given§ 9, (JT) == o mod i;. 1=1
The faithful representation of W on t is generated by the reflections of t in the planes u_i of D'(G).
“(9.6)The faithful representation of W on t is generated by the reflections of t in the planes VLi of D'(G);”
Quote in the paperNot a substantial claim - Premisenone given§ 9, (JT) == o mod i;. 1=1
Every fundamental chamber of D'(G) is simultaneously a fundamental domain of W in t.
“(9.7) Every fundamental chamber, ^, oi D'(G) is simultaneously a fundamental domain of W in t.”
Quote in the paperNot a substantial claim - Premisenone given§ 9, (JT) == o mod i;. 1=1
If G is connected, simply connected and compact, W is described by D(G).
“If Gis connected, simply connected and compact, TV is described by D(G).”
Quote in the paperSubstantial claim - Interpretationreasoning§ o'ec'
Because the indices are even, the Morse inequalities must be actual equalities for purely dimensional reasons.
“Because of this very fortuitous circumstance the Morse inequalities must be actual equalities for purely « dimensional » reasons.”
Quote in the paperSubstantial claim - Premisecitation§ 9. THE DIAGRAM OF G.
Ui has at most two components.
“(9.4) Ui has at most two components,”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [I]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [7] 7. H. HOPF, Maximale Toroide und singulare elemente in geschlossenen Lieschen gruppen (Corn. Math. ffelv., t. 15, 1942, p. 59) source
- cites [12] 12. J.-P. SERRE, b. Groupes d'homotopie et classes de groupes abeliens {Ann* Mat^ t. 58, i953, p. 258).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Premisecitation§ 9. THE DIAGRAM OF G.
dim u_i ∩ u_j = l-2 if i≠j.
“(9.3) dim^n^/:=^-2 if i^j.”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [I]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [7] 7. H. HOPF, Maximale Toroide und singulare elemente in geschlossenen Lieschen gruppen (Corn. Math. ffelv., t. 15, 1942, p. 59) source
- cites [12] 12. J.-P. SERRE, b. Groupes d'homotopie et classes de groupes abeliens {Ann* Mat^ t. 58, i953, p. 258).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Premisenone given§ 9, (JT) == o mod i;. 1=1
W is generated by the reflections in the planes of D(G).
“(9.8) W is generated by the reflections in the planes ofD(G).”
Quote in the paperSubstantial claim
- It may state two separate claims as one. 4
- Probably not a substantial claim. 3
- The quote only partly states this claim. 5
It may state two separate claims as one.
Findingreasoning§ 8. THE GEODES1CS OF (G, TV, P).Every segment s in S(G,N,P) is perpendicular to o(P) at its end point and tangent to T at P; hence s lies in T.
“Hence every segment s<E S (G, 7V, P) (which must be perpendicular to the orbit N at its start) is perpendicular to o (P) at its end point.The segment s is therefore tangent to T at P. Because T is a subgroup, s will have to lie in T.”
Quote in the paperSubstantial claimProbably not a substantial claim.
Findingnone given§ 9, (JT) == o mod i;. 1=1The extended Weyl group W is generated by W and the covering transformations of t→T.
“This^ is the transformation group W of t, generated by W and the covering transformations of i-> T.”
Quote in the paperSubstantial claimProbably not a substantial claim.
Findingreasoning§ j--\D'(G) consists of the points which are in the kernel of at least one of the root forms θ_i.
“In any case D' {G) consists of the points which are in the kernel of at least one of the 9^.”
Quote in the paperSubstantial claimThe quote only partly states this claim.
Findingreasoning§ 8. THE GEODES1CS OF (G, TV, P).Conversely, a geodesic segment on T joining T' in N∩T to P is in S(M,N,P).
“Conversely a geodesic segment on T joining T'eA'nT 7 lo P, is perpendicular to o(r) at its initial point and therefore is in S(M, A, P).”
Quote in the paperSubstantial claimThe quote only partly states this claim.
Findingnone given§ 9, (JT) == o mod i;. 1=1The lattice Γ is left invariant by the action of W.
“The lattice T is left invariant by the action of W.”
Quote in the paperSubstantial claimThe quote only partly states this claim.
Findingnone given§ --i /'For a compact connected simply-connected Lie group G, a maximal torus T, an orbit N of the adjoint action, and a general point P of T not on N, the space of paths Ω(G,N,P) is free of torsion.
“THEOREM I. -The space of paths, ^l (G, N, P), is free of torsion.”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
- The authors state this as an assumption: T is a maximal torus of G.
- The authors state this as an assumption: N is an orbit of the adjoint action of G on G.
- The authors state this as an assumption: P is a general point of T not on N.
It may state two separate claims as one.
Findingnone given§ 9. THE DIAGRAM OF G.The adjoint action of T on g decomposes g into orthogonal invariant subspaces g = t ⊕ e1 ⊕ e2 ⊕ ... ⊕ em, with t pointwise fixed under Ad|T, while each ei is a two-plane on which T is represented nontrivially by rotations.
“This action of T on 9 decomposes q into orthogonal invariant subspaces, (^i) 9= : t®el®e,©...®e^ with t poinlwise fixed under Ad | 7 7 , while each Oi is a two-plane on which T is represented nonlrivially by rotations.”
Quote in the paperSubstantial claimProbably not a substantial claim. The paper’s conclusions may not depend on it.
Findingnone given§ 9, (JT) == o mod i;. 1=1A fundamental chamber singles out precisely m root forms which take positive values in it.
“A fundamental chamber 5^, singles out precisely m-root forms which take on positive values in ^F.”
Quote in the paperNot a substantial claimThe quote only partly states this claim.
Findingreasoning§ Additional informationThe function space Ω(M,N,P) is of the same homotopy type as N.
“PROPOSITION.11.5.-The function space i2(Af, TV, P) is of the same homotopy type as TV.”
Quote in the paperSubstantial claim- The authors state this as an assumption: M is an open spherical disc about the origin in g, so small that exp|M maps M homeomorphically into G.
It may state two separate claims as one.
Findingreasoning§ --i /'Let s be a transversal segment on G. Then: (a) every element V∈J^ is of the form V=π_*(Y)+ω_*(Z), with Y∈g, Z∈c(s); (b) if for V∈J^, Z∈c(s), π_*(Y)+ω_*(Z)=0.
“LEMMA 7 . 4-. -Let s be a transversal segment on G. Then : (a) Every element V ^.J^ is of the form r==7r.,(r)+c*),s.(Z),[reg, Zec(^)]; (b) If for Fe^ Z^c(s), 7T,(r)+CO,(Z)=0.”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
The quote only partly states this claim.
Findingreasoning§ --i /'The Poincare series of Ω(G,N,P) coincides with the Morse series of (G,N,P).
“Its odd Betti numbers vanish, and the Poincare series of Sl(G, N, P) coincides with the Morse series of (G, TV, P).”
Quote in the paperSubstantial claim- The authors state this as an assumption: G is a compact connected and simply-connected Lie group.
- The authors state this as an assumption: T is a maximal torus of G.
- The authors state this as an assumption: N is an orbit of the adjoint action of G on G.
- The authors state this as an assumption: P is a general point of T not on N.
It may state two separate claims as one.
Premisereasoning§ 9, (JT) == o mod i;. 1=1The closure of every fundamental cell is compact and contains precisely one point of the lattice Γ.
“(9.io)The closure of every fundamental cell is compact and contains precisely one point of the lattice r.”
Quote in the paperSubstantial claim
The authors state this as an assumption.
G is a compact connected and simply-connected Lie group.
“Let G be a compact connected and simply-connected Lie group, TC.G a maximal torus of G, N any orbit of the adjoint action of G on G.”
The authors state this as an assumption.
T is a maximal torus of G.
“Let G be a compact connected and simply-connected Lie group, TC.G a maximal torus of G, N any orbit of the adjoint action of G on G.”
The authors state this as an assumption.
N is an orbit of the adjoint action of G on G.
“Let G be a compact connected and simply-connected Lie group, TC.G a maximal torus of G, N any orbit of the adjoint action of G on G.”
The authors state this as an assumption.
P is a general point of T not on N.
“Let P be a general point of T not on N.”
The authors state this as an assumption.
M is an open spherical disc about the origin in g, so small that exp|M maps M homeomorphically into G.
“Let Mbe an open, spherical disc about the origin in $, which is so small that exp | Sl maps Sl homeomorphically into 6r.”
The authors state this as an assumption.
G is simple, connected, simply connected and compact.
“Let G be simple^ connected^ simply connected and compact.”
The authors state this as an assumption.
The Morse series of (M,N,P) exists, with P a regular point of M-N.
“Suppose then that the Morse series of (M, N, P) exists (P a regular point of M-N).”
The authors state this as an assumption.
E∞ is of finite type.
“Suppose now that E^ is of finite type so that the above proposition applies.”
Citations
Cites
Homologie Singuliere Des Espaces Fibres
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The Calculus of Variations in the Large
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THE STABLE HOMOTOPY OF THE CLASSICAL GROUPS
Bott, R. (1957) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.43.10.933 · Cited by 80
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Self-citationMaximale Toroide und singuläre Elemente in geschlossenen Lieschen Gruppen
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Flagged first, then by citations
Research works printed but not linked: 2
- [2]2. C. CHEVALLEY, Theory of Lie groups I (Princeton, New Jersey, Princeton University press, 1946).
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Cited by
Sorted by citations
Marsh, D. J. E. (2016) · Physics Reports · doi:10.1016/j.physrep.2016.06.005 · Cited by 2,305
Self-duality in four-dimensional Riemannian geometry
Atiyah, M. F., Hitchin, N. J., & Singer, I. M. (1978) · Proceedings of the Royal Society of London A Mathematical and Physical Sciences · doi:10.1098/rspa.1978.0143 · Cited by 1,489
Chiral rings in N = 2 superconformal theories
Lerche, W., Vafa, C., & Warner, N. P. (1989) · Nuclear Physics B · doi:10.1016/0550-3213(89)90474-4 · Cited by 905
On some Bruhat decomposition and the structure of the Hecke rings of p -adic Chevalley groups
Iwahori, N., & Matsumoto, H. (1965) · Publications mathématiques de l IHÉS · doi:10.1007/bf02684396 · Cited by 532
“X((r) is nothing but the function considered /">»^ by Bott [2].”
Coleman, S. (1979) · The Whys of Subnuclear Physics · doi:10.1007/978-1-4684-0991-8_16 · Cited by 445
Fermion number fractionization in quantum field theory
Niemi, A. J., & Semenoff, G. W. (1986) · Physics Reports · doi:10.1016/0370-1573(86)90167-5 · Cited by 429
Extrema of mass, stationarity, and staticity, and solutions to the Einstein-Yang-Mills equations
Sudarsky, D., & Wald, R. M. (1992) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.46.1453 · Cited by 259
Self-dual Yang-Mills connections on non-self-dual 4-manifolds
Taubes, C. H. (1982) · Journal of Differential Geometry · doi:10.4310/jdg/1214436701 · Cited by 227
“The manifold M has cells up to dimension 4, whereas the 4-skeleton of BG is homotopically a disjoint union of / 4-spheres, [6], [7], Every map from M into BG is homotopic to a map of M into the 4-skeleton [26].”
BackgroundBrowder, W. (1961) · Annals of Mathematics · doi:10.2307/1970305 · Cited by 202
New examples of manifolds with strictly positive curvature
Eschenburg, J.-H. (1982) · Inventiones mathematicae · doi:10.1007/bf01389224 · Cited by 201
“Moreover, it is known for any compact Lie group that n 2 = 0 and n3=Z k where k is the number of simple factors of the Lie algebra [4].”
BackgroundSous-groupes commutatifs et torsion des groupes de Lie compacts connexes
Borel, A. (1961) · Tohoku Mathematical Journal · doi:10.2748/tmj/1178244298 · Cited by 169
A Survey on Isoparametric Hypersurfaces and Their Generalizations
Thorbergsson, G. (2000) · Handbook of differential geometry · doi:10.1016/s1874-5741(00)80013-8 · Cited by 170
A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle
Hermann, R. B. (1960) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1960-0112151-4 · Cited by 156
The Poincar� series of a Coxeter group
MacDonald, I. G. (1972) · Mathematische Annalen · doi:10.1007/bf01431421 · Cited by 147
Spectra of Wess-Zumino-Witten models with arbitrary simple groups
Felder, G., Gawędzki, K., & Kupiainen, A. J. (1988) · Communications in Mathematical Physics · doi:10.1007/bf01228414 · Cited by 141
Yang's System of Particles and Hecke Algebras
Heckman, G., & Opdam, E. (1997) · Annals of Mathematics · doi:10.2307/2951825 · Cited by 125
“22 will give the explicit formula of Bott and Macdonald for the Poincare series of affine Weyl groups ([3], [23]).”
MethodFreed, D. S. (1988) · Journal of Differential Geometry · doi:10.4310/jdg/1214442279 · Cited by 124
“First we recall that the cohomology of QG is torsion-free and is concentrated in even dimensions [11].”
BackgroundMorse-Bott theory and equivariant cohomology
Austin, D. E., & Braam, P. J. (1995) · Progress in mathematics · doi:10.1007/978-3-0348-9217-9_8 · Cited by 114
Bousquet‐Mélou, M., & Eriksson, K. (1997) · The Ramanujan Journal · doi:10.1023/a:1009771306380 · Cited by 101
“Bott [ 2 ] gave, as an application of Morse theory to the topology of Lie groups, a general formula for the Poincar´ e series of the affine groups in terms of the Poincar´ e series of the finite groups and their “exponents”.”
MethodRegular Functions on Certain Infinite-dimensional Groups
Kac, V. G., & Peterson, D. H. (1983) · Birkhäuser Boston eBooks · doi:10.1007/978-1-4757-9286-7_8 · Cited by 100
On the Differential Geometry of Foliations
Hermann, R. B. (1960) · Annals of Mathematics · doi:10.2307/1970226 · Cited by 98
Almost commuting elements in compact Lie groups
Borel, A., Friedman, R. S., & Morgan, J. W. (2002) · Memoirs of the American Mathematical Society · doi:10.1090/memo/0747 · Cited by 95
Harmonic two-spheres in compact symmetric spaces, revisited
Burstall, F. E., & Guest, M. A. (1997) · Mathematische Annalen · doi:10.1007/s002080050127 · Cited by 95
Generalized Verma modules, loop space cohomology and MacDonald-type identities
Lepowsky, J. (1979) · Annales Scientifiques de l École Normale Supérieure · doi:10.24033/asens.1365 · Cited by 95
Browder, W. (1963) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1963-0145530-7 · Cited by 95
“We recall that it is a theorem of Bott [4] that if G is a Lie group, then //^(QG) has no torsion.”
BackgroundPolar actions on rank-one symmetric spaces
Podestà, F., & Thorbergsson, G. (1999) · Journal of Differential Geometry · doi:10.4310/jdg/1214425449 · Cited by 89
ON THE SPECTRAL DECOMPOSITION OF AFFINE HECKE ALGEBRAS
Opdam, E. (2004) · Journal of the Institute of Mathematics of Jussieu · doi:10.1017/s1474748004000155 · Cited by 77
“This result was first (for equal labels, using Morse theory) derived by Bott [9], and by elementary means by Steinberg [47].”
MethodMinimum and Conjugate Points in Symmetric Spaces
Crittenden, R. B. (1962) · Canadian Journal of Mathematics · doi:10.4153/cjm-1962-024-8 · Cited by 77
“T h e above is a s ta tement of the variational completeness of the action of Hon M (2; 4) .”
BackgroundThe classification of p-compact groups for p odd
Andersen, K. K. S., Grodal, J., Møller, J., & Viruel, A. (2008) · Annals of Mathematics · doi:10.4007/annals.2008.167.95 · Cited by 78
“The centralizer decomposition theorem of Dwyer-Wilkerson [57, Thm. 8.1], generalizing a theorem for compact Lie groups by Jackowski-McClure [81, Thm. 1.3], says that the evaluation map hocolimA(X) BCX → BX induces an isomorphism on mod p homology.”
BackgroundInfluentialPowers of the Euler product and commutative subalgebras of a complex simple Lie algebra
Kostant, B. (2004) · Inventiones mathematicae · doi:10.1007/s00222-004-0370-7 · Cited by 72
“ar´e polynomial of K. This makes mi here have value 1 less than the value assigned to mi in [B]. The following is a classic result of Bott on the Poincar´e series of Ω(K). See Theorem B and (13.2) in [B]. Theorem 1.6 (Bott). The Poincar´e series of the loop group Ω(K) is P(t2). Furthermore P(t) = Y i∈I 1/(1−tmi) (1.15…”
BackgroundInfluentialMimura, M. (1995) · Elsevier eBooks · doi:10.1016/b978-044481779-2/50020-1 · Cited by 65
Intersection cohomology methods in representation theory
Lusztig, G. (1990) · Iwanami eBooks · Cited by 67
Constructing Groups Associated to Infinite-Dimensional Lie Algebras
Kac, V. G. (1985) · Mathematical Sciences Research Institute publications · doi:10.1007/978-1-4612-1104-4_7 · Cited by 67
Locally symmetric homogeneous spaces
Wolf, J. A. (1962) · Commentarii Mathematici Helvetici · doi:10.1007/bf02566963 · Cited by 63
“Orthogonal to some ad (G) orbit on G, a is orthogonal to every ad (Cr)-orbit, whence cr' (1) is tangent to the centralizer of hvh~x [5].”
BackgroundA Sufficient Condition that a Mapping of Riemannian Manifolds be a Fibre Bundle
Hermann, R. B. (1960) · Proceedings of the American Mathematical Society · doi:10.2307/2032963 · Cited by 63
On the motivic class of the stack of bundles
Behrend, K. A., & Dhillon, A. (2006) · Advances in Mathematics · doi:10.1016/j.aim.2006.11.003 · Cited by 61
Curvature explosion in quotients and applications
Lytchak, A., & Thorbergsson, G. (2010) · Journal of Differential Geometry · doi:10.4310/jdg/1284557927 · Cited by 57
“Variationally complete actions were introduced by Bott and used in [Bot56] and [BS58], as a mean for studying the topology of (loop spaces of) symmetric spaces.”
MethodGroup actions and deformations for harmonic maps
Guest, M. A., & Ohnita, Y. (1993) · Journal of the Mathematical Society of Japan · doi:10.2969/jmsj/04540671 · Cited by 57
“up C(TQ) = Ul× Un−l, and hence is a copy of Grw 0 (Cl) × Grw 1 (Cn−l), where w i = dim Wi. The index of a critical manifold may be computed using the Stiefel diagram of G. This theory is due to Bott ([Bo]). Let ∇hQbe the gradient of hQwith respect to the Kahler metric. The integral curves of ∇hQmay be calculated expli…”
MethodPolar actions on symmetric spaces
Kollross, A. (2007) · Journal of Differential Geometry · doi:10.4310/jdg/1193074901 · Cited by 55
“Bott [5] and Bott and Samelson [6] considered the adjoint action of a compact Lie group on itself and on its Lie algebra [5] and more generally, the isotropy action of a compact symmetric space [6].”
BackgroundThe group of isometries of a left invariant Riemannian metric on a Lie group
Ochiai, T., & Takahashi, T. (1976) · Mathematische Annalen · doi:10.1007/bf01360280 · Cited by 54
Browder, W. (1963) · Transactions of the American Mathematical Society · doi:10.2307/1993874 · Cited by 54
Affine Permutations of Type $A$
Björner, A., & Brenti, F. (1995) · The Electronic Journal of Combinatorics · doi:10.37236/1276 · Cited by 52
“6 we obtain the following expression for the length-generating function of Ãn−1, which is the type A specialization of a formula by Bott [3], (see also [15], §8.”
BackgroundInfluentialAffine Weyl Groups as Infinite Permutations
Eriksson, H., & Eriksson, K. (1998) · The Electronic Journal of Combinatorics · doi:10.37236/1356 · Cited by 49
“Finally, as an application of our theory, we give new combinatorial proofs of Bott’s formulas for the Poincaré series of the affine groups Ãn, B̃n, C̃n, D̃n.”
MethodVariational completeness for compact symmetric spaces
Hermann, R. B. (1960) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1960-0124436-6 · Cited by 47
“Bott and Samelson have proved the following: (1) L acts in a variationally complete manner on G/L, the right coset space, (2) L acts via the linear isotropy representation in a variationally complete way on the tangent space at a point of G/L.”
BackgroundA filtration of the loops onSU(N) by Schubert varieties
Mitchell, S. Α. (1986) · Mathematische Zeitschrift · doi:10.1007/bf01229802 · Cited by 44
Geometric resolution of singular Riemannian foliations
Lytchak, A. (2010) · Geometriae Dedicata · doi:10.1007/s10711-010-9488-5 · Cited by 42
“Since singular Riemannian foliations without horizontal conjugate points generalize the concept of variationally complete actions introduced in [Bot56] and [BS58] and investigated in [Con72],[GT02], [DO01] and [LT07b], the next result also gives a description of variationally complete actions in terms of the quotient…”
BackgroundA Bruhat decomposition for the loop space of a compact group: A new approach to results of Bott
Garland, H., & Raghunathan, M. S. (1975) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.72.12.4716 · Cited by 41
Hopf algebras over Dedekind domains and torsion inH-spaces
Clark, A. (1965) · Pacific Journal of Mathematics · doi:10.2140/pjm.1965.15.419 · Cited by 40
The periodicity theorem for the classical groups and some of its applications
Bott, R. (1970) · Advances in Mathematics · doi:10.1016/0001-8708(70)90030-7 · Cited by 38
Self-citationKilling the middle homotopy groups of odd dimensional manifolds
Wall, C. T. C. (1962) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1962-0139185-4 · Cited by 38
Universal p-adic Zeta Functions and Their Functional Equations
Igusa, J.-I. (1989) · American Journal of Mathematics · doi:10.2307/2374877 · Cited by 35
Rigidity properties of compact Lie groups modulo maximal tori
Papadima, Ş. (1986) · Mathematische Annalen · doi:10.1007/bf01459142 · Cited by 32
“[ 5 ]), extension of scalars identifies AutH*(G/T; Z) with a certain subgroup of AutH*(G/T; R), which is plainly contained in the subgroup of those elements of AutH*(G/T; R) which induce unimodular self-maps of H2(G/T; 7.).”
BackgroundFloer homology of families IPreprint
Hutchings, M. L. (2003) · Cited by 31
Yang-Mills connections on nonorientable surfaces
Ho, N.-K., & Liu, C.-C. M. (2008) · Communications in Analysis and Geometry · doi:10.4310/cag.2008.v16.n3.a6 · Cited by 28
Dedekind's η-function and the cohomology of infinite dimensional Lie algebras
Garland, H. (1975) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.72.7.2493 · Cited by 29
The Lefschetz thimble and the sign problem
Scorzato, L. (2016) · Proceedings of The 33rd International Symposium on Lattice Field Theory — PoS(LATTICE 2015) · doi:10.22323/1.251.0016 · Cited by 27
Progress in the theory of singular Riemannian foliations
Alexandrino, M. M., Briquet, R., & Töben, D. (2013) · Differential Geometry and its Applications · doi:10.1016/j.difgeo.2013.01.004 · Cited by 27
“Bott originally introduced variational completeness in [19], and with Samelson in [20].”
Variationally complete actions on nonnegatively curved manifolds
Lytchak, A., & Thorbergsson, G. (2007) · Illinois Journal of Mathematics · doi:10.1215/ijm/1258138433 · Cited by 27
“Variationally complete actions were first introduced by Bott in [2] and later in a slightly different way in [3].”
BackgroundThe growth function of a discrete group
Wagreich, P. (1982) · Lecture notes in mathematics · doi:10.1007/bfb0101514 · Cited by 27
What Chern–Simons theory assigns to a point
Henriques, A. (2017) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1711591114 · Cited by 45
“If G is simple and simply connected, there is a canonical isomorphism H4(BG,Z) ∼= Z [17].”
BackgroundDiscrete subgroups of lie groups
Mostow, G. D. (1975) · Advances in Mathematics · doi:10.1016/0001-8708(75)90103-6 · Cited by 26
The integral cohomology ring of the symmetric space EVII
Watanabe, T. (1975) · Kyoto journal of mathematics · doi:10.1215/kjm/1250523069 · Cited by 26
Hutchings, M. (2008) · Algebraic & Geometric Topology · doi:10.2140/agt.2008.8.435 · Cited by 25
String topology for Lie groups
Hepworth, R. (2010) · Journal of Topology · doi:10.1112/jtopol/jtq012 · Cited by 24
“It follows from Bott’s construction that the Hopf algebra morphism H ∗ (Ω 0 SO (2 m + 1); Q ) → H ∗ (Ω 0 SO (2 m + 2); Q ) is given by σ i 7→ σ i for i = 0 , . . . , 2 m − 1 and that H ∗ (Ω 0 SO (2 m + 2); Q ) → H ∗ (Ω 0 SO (2 m + 3); Q ) is given by ε 7→ 0 and σ i 7→ σ i for i = 0 , . . . , 2 m .”
BackgroundInfluentialThe integral cohomology ring of $E_7/T$
Nakagawa, M. (2001) · Kyoto journal of mathematics · doi:10.1215/kjm/1250517635 · Cited by 24
D’Ariano, G. M. (1985) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/18/9/013 · Cited by 24
The adjoint action of a Lie group on the space of loops
Kono, A., & Kozima, K. (1993) · Journal of the Mathematical Society of Japan · doi:10.2969/jmsj/04530495 · Cited by 22
Integral conjugacy classes of compact Lie groups
Mohrdieck, S., & Wendt, R. (2004) · manuscripta mathematica · doi:10.1007/s00229-004-0447-z · Cited by 22
Vanishing of one-dimensional L 2 -cohomologies of loop groups
Aida, S. (2011) · Journal of Functional Analysis · doi:10.1016/j.jfa.2011.06.003 · Cited by 21
Topology and physics-a historical essayPreprint
Nash, C. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.hep-th/9709135 · Cited by 21
Cohomology modulo $2$ of the compact exceptional groups $E_6$ and $E_7$
Araki, S. (1961) · Osaka City University (Osaka City University) · Cited by 21
Copolarity of isometric actions
Gorodski, C., Olmos, C. E., & Tojeiro, R. (2003) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-03-03427-5 · Cited by 20
“In particular, we develop some of the structural theory of copolarity k representations, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Samelson.”
MethodProblems of growth and rationality in algebra and topology
Babenko, I. K. (1986) · Russian Mathematical Surveys · doi:10.1070/rm1986v041n02abeh003243 · Cited by 20
On the Non-Commutativity of Pontrjagin Rings Mod 3 of Some Compact Exceptional Groups
Araki, S. (1960) · Nagoya Mathematical Journal · doi:10.1017/s0027763000002166 · Cited by 18
Representations of compact Lie groups and the osculating spaces of their orbitsPreprint
Gorodski, C., & Thorbergsson, G. (2002) · arXiv (Cornell University) · doi:10.48550/arxiv.math/0203196 · Cited by 19
“The action of G on M is called variationally complete if every G-transversal Jacobi field J in M that is tangent to the Gorbits at two different parameter values is the restriction of a Killing field on M induced by the G-action (see [4] and [5]).”
Descent Identities, Hessenberg Varieties, and the Weil Conjectures
Fulman, J. (1999) · Journal of Combinatorial Theory Series A · doi:10.1006/jcta.1999.2964 · Cited by 19
“Chevalley [2] and Bott [1] used the topology of compact Lie groups to prove the factorization of the Poincaré series for Weyl groups.”
MethodThe image of H∗(BG,Z) in H∗(BT,Z) for G a compact lie group with maximal torus T
Feshbach, M. (1981) · Topology · doi:10.1016/0040-9383(81)90015-x · Cited by 19
The Topology of Certain Spaces of Paths on a Compact Symmetric Space
Conlon, L. (1964) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1964-0163328-1 · Cited by 20
On the construction of perfect Morse functions on compact manifolds of coherent states
Berceanu, S., & Gheorghe, A. H. (1987) · Journal of Mathematical Physics · doi:10.1063/1.527691 · Cited by 18
Flag Manifolds and Infinite Dimensional Kähler Geometry
Freed, D. S. (1985) · Mathematical Sciences Research Institute publications · doi:10.1007/978-1-4612-1104-4_3 · Cited by 18
Variational Completeness for Compact Symmetric Spaces
Hermann, R. B. (1960) · Proceedings of the American Mathematical Society · doi:10.2307/2034708 · Cited by 18
Gravitational lensing by dark matter caustics
Charmousis, C., Onemli, V. K., Qiu, Z., et al. (2003) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.67.103502 · Cited by 17
“A theorem due to Bott [68] states that any continuous mapping of S(3) into a simple Lie group G can be continuously deformed to a mapping into an SU(2) subgroup of G.”
BackgroundTaut Immersions into Complete Riemannian Manifolds
Terng, C.-L., & Thorbergsson, G. (1997) · doi:10.1017/9781009701815.006 · Cited by 17
Variationally Complete Actions on Compact Symmetric Spaces
Gorodski, C., & Thorbergsson, G. (2002) · Journal of Differential Geometry · doi:10.4310/jdg/1090425528 · Cited by 16
“Variationally complete actions were introduced by Bott in [1] (see also [2]).”
BackgroundTopological Charges in Gauge Theories
Leinaas, J. M. (1980) · Fortschritte der Physik · doi:10.1002/prop.19800281102 · Cited by 16
Affine Partitions and Affine Grassmannians
Billey, S., & Mitchell, S. Α. (2009) · The Electronic Journal of Combinatorics · doi:10.37236/84 · Cited by 15
“Compare the result with Bott’s formula.”
BackgroundInfluentialDuistermaat, J. J., & Kolk, J. A. C. (2000) · Universitext · doi:10.1007/978-3-642-56936-4_1 · Cited by 16
Topology and Physics – a Historical Essay
Nash, C. (1999) · Elsevier eBooks · doi:10.1016/b978-044482375-5/50013-4 · Cited by 37
Factorizations and induced homomorphisms
Olum, P. (1969) · Advances in Mathematics · doi:10.1016/0001-8708(69)90003-6 · Cited by 15
Commutative subgroups and torsion in compact Lie groups
Borel, A. (1960) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1960-10474-0 · Cited by 15
“G/Z(S) is torsion free when S is a torus [5; 10 ], and the fact that if G is simply connected, the centralizer Z(g) of any element of g is connected, which can be proved using Proposition 2, p. 56 of [13] and some properties of singular elements.”
BackgroundVariationally complete representations are polar
Scala, A. J. D., & Olmos, C. E. (2001) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-01-06226-8 · Cited by 14
A Weyl group generating function that ought to be better known
Stembridge, J. R., & Waugh, D. J. (1998) · Indagationes Mathematicae · doi:10.1016/s0019-3577(98)80012-8 · Cited by 14
Quantized charge transport, chiral anomalies, and fractional charge
Zee, A. (1984) · Physics Letters B · doi:10.1016/0370-2693(84)90397-6 · Cited by 14
The Hyperbolic Yang–Mills Equation for Connections in an Arbitrary Topological Class
Oh, S., & Tataru, D. (2019) · eScholarship (California Digital Library) · Cited by 13
Smooth and palindromic Schubert varieties in affine Grassmannians
Billey, S., & Mitchell, S. Α. (2009) · Journal of Algebraic Combinatorics · doi:10.1007/s10801-009-0181-4 · Cited by 13
“Theorem 3.1 [ 4 ] Let e1, e2, . . . , en be the exponents of W. Then”
BackgroundInfluentialLoop group actions on harmonic maps and their applications
Guest, M. A., & Ohnita, Y. (1994) · Aspects of mathematics. E · doi:10.1007/978-3-663-14092-4_12 · Cited by 13
The bott filtration of a loop group
Mitchell, S. Α. (1987) · Lecture notes in mathematics · doi:10.1007/bfb0083012 · Cited by 13
“Bott, An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France 84(1956), 251-282.”
MethodKilling the Middle Homotopy Groups of Odd Dimensional Manifolds
Wall, C. T. C. (1962) · Transactions of the American Mathematical Society · doi:10.2307/1993837 · Cited by 13
Rot, T. O., & Vandervorst, R. (2014) · Journal of Topology and Analysis · doi:10.1142/s1793525314500174 · Cited by 12
Dynamic homotopy and landscape dynamical set topology in quantum control
Dominy, J., & RABITZ, H. A. (2012) · Journal of Mathematical Physics · doi:10.1063/1.4742375 · Cited by 12
“The integral cohomologies of the complex Grassmannian manifolds Gr ν n and of the loop space of the special unitary group ΩSU( N ) are free of torsion and zero for odd dimension [33] as desired for the sim-plification indicated above.”
Metadata
- Authors
- Bott, R.
- Corresponding author
- Bott, Raoul
- Venue
- Bulletin de la Société mathématique de France
- Publisher
- Societe Mathematique de France
- Published
- 1956
- Volume
- 79
- Pages
- 251-281
- ISSN
- 0037-9484 (ISSN-L)
- 2102-622X
- OpenAlex
- W35421105
- Cited by
- 280
- References
- 5
- Citations by year
- 2026: 4 · 2025: 5 · 2024: 5 · 2023: 3 · 2022: 7 · 2021: 3 · 2020: 5 · 2019: 4 · 2018: 4 · 2017: 17 · 2016: 9 · 2015: 6 · 2014: 5 · 2013: 6 · 2012: 6
- FWCI
- 0.00
- Type
- journal article
- Language
- zh
- Open access
- bronze
- doi.org
- Locations
- Bulletin de la Société mathématique de France (publishedVersion)
- French digital mathematics library (Numdam) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Algebra and Number Theory › Advanced Algebra and Geometry
- Keywords
- Morse theory, compact Lie groups, loop spaces, Poincaré series, Betti numbers, root form, Weyl group, simply connected Lie groups, free abelian groups, symmetric spaces
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 2026-10-11, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-11 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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