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Ken
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Recall that Bott's obstruction for integrability [Bott70] asserts that:

Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$.

In [p. 144, Hae71], Haefliger says:

($\ast$) The theorem of Bott ... implies that the induced homomorphism $$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$ is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r\geq 1$$r> 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.)

Can anyone explain why Bott's theorem implies ($\ast$)? I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations.

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

P.S. In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in bijection with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.

References

[Bott70] Bott, Raoul. On a topological obstruction to integrability. Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968), pp. 127–131 Proc. Sympos. Pure Math., XIV-XVI American Mathematical Society, Providence, RI, 1970

[Hae71] Haefliger. A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163 Lecture Notes in Math., Vol. 197 Springer-Verlag, Berlin-New York, 1971

[BH71] Bott. R; Haefliger. A, On characteristic classes of $\Gamma$-foliations. Bull. Amer. Math. Soc.78(1972), 1039–1044.

Recall that Bott's obstruction for integrability [Bott70] asserts that:

Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$.

In [p. 144, Hae71], Haefliger says:

($\ast$) The theorem of Bott ... implies that the induced homomorphism $$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$ is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r\geq 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.)

Can anyone explain why Bott's theorem implies ($\ast$)? I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations.

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

P.S. In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in bijection with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.

References

[Bott70] Bott, Raoul. On a topological obstruction to integrability. Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968), pp. 127–131 Proc. Sympos. Pure Math., XIV-XVI American Mathematical Society, Providence, RI, 1970

[Hae71] Haefliger. A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163 Lecture Notes in Math., Vol. 197 Springer-Verlag, Berlin-New York, 1971

[BH71] Bott. R; Haefliger. A, On characteristic classes of $\Gamma$-foliations. Bull. Amer. Math. Soc.78(1972), 1039–1044.

Recall that Bott's obstruction for integrability [Bott70] asserts that:

Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$.

In [p. 144, Hae71], Haefliger says:

($\ast$) The theorem of Bott ... implies that the induced homomorphism $$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$ is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r> 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.)

Can anyone explain why Bott's theorem implies ($\ast$)? I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations.

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

P.S. In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in bijection with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.

References

[Bott70] Bott, Raoul. On a topological obstruction to integrability. Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968), pp. 127–131 Proc. Sympos. Pure Math., XIV-XVI American Mathematical Society, Providence, RI, 1970

[Hae71] Haefliger. A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163 Lecture Notes in Math., Vol. 197 Springer-Verlag, Berlin-New York, 1971

[BH71] Bott. R; Haefliger. A, On characteristic classes of $\Gamma$-foliations. Bull. Amer. Math. Soc.78(1972), 1039–1044.

updated the reference
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Ken
Ken
  • 2.3k
  • 10
  • 19

Recall that Bott's obstruction for integrability [Bott71][Bott70] asserts that:

Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$.

In [p. 144, Hae71], Haefliger says:

($\ast$) The theorem of Bott ... implies that the induced homomorphism $$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$ is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r\geq 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.)

Can anyone explain why Bott's theorem implies ($\ast$)? I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations.

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

P.S. In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in bijection with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.

References

[Bott71][Bott70] Bott. R, Raoul. On a topological obstructionsobstruction to integrability. Actes du Congrès International des MathématiciensGlobal Analysis (NiceProc. Sympos. Pure Math., 1970)Vols. XIV, Tome 1XV, XVI, Berkeley, Calif., 1968), pp. 27–36127–131 GauthierProc. Sympos. Pure Math., XIV-Villars ÉditeurXVI American Mathematical Society, ParisProvidence, 1971RI, 1970

[Hae71] Haefliger. A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163 Lecture Notes in Math., Vol. 197 Springer-Verlag, Berlin-New York, 1971

[BH71] Bott. R; Haefliger. A, On characteristic classes of $\Gamma$-foliations. Bull. Amer. Math. Soc.78(1972), 1039–1044.

Recall that Bott's obstruction for integrability [Bott71] asserts that:

Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$.

In [p. 144, Hae71], Haefliger says:

($\ast$) The theorem of Bott ... implies that the induced homomorphism $$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$ is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r\geq 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.)

Can anyone explain why Bott's theorem implies ($\ast$)? I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations.

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

P.S. In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in bijection with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.

References

[Bott71] Bott. R, On topological obstructions to integrability. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 27–36 Gauthier-Villars Éditeur, Paris, 1971

[Hae71] Haefliger. A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163 Lecture Notes in Math., Vol. 197 Springer-Verlag, Berlin-New York, 1971

[BH71] Bott. R; Haefliger. A, On characteristic classes of $\Gamma$-foliations. Bull. Amer. Math. Soc.78(1972), 1039–1044.

Recall that Bott's obstruction for integrability [Bott70] asserts that:

Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$.

In [p. 144, Hae71], Haefliger says:

($\ast$) The theorem of Bott ... implies that the induced homomorphism $$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$ is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r\geq 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.)

Can anyone explain why Bott's theorem implies ($\ast$)? I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations.

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

P.S. In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in bijection with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.

References

[Bott70] Bott, Raoul. On a topological obstruction to integrability. Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968), pp. 127–131 Proc. Sympos. Pure Math., XIV-XVI American Mathematical Society, Providence, RI, 1970

[Hae71] Haefliger. A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163 Lecture Notes in Math., Vol. 197 Springer-Verlag, Berlin-New York, 1971

[BH71] Bott. R; Haefliger. A, On characteristic classes of $\Gamma$-foliations. Bull. Amer. Math. Soc.78(1972), 1039–1044.

updated the reference
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Ken
Ken
  • 2.3k
  • 10
  • 19

Recall that Bott's obstruction for integrability [Bott71] asserts that:

Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$.

In [p. 144, Hae71], Haefliger says:

($\ast$) The theorem of Bott ... implies that the induced homomorphism $$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$ is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r\geq 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.)

Can anyone explain why Bott's theorem implies ($\ast$)? I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations.

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

P.S. In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in bijection with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.

References

[Bott71] Bott. R, On topological obstructions to integrability. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 27–36 Gauthier-Villars Éditeur, Paris, 1971

[Hae71] Haefliger,. A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163 Lecture Notes in Math., Vol. 197 Springer-Verlag, Berlin-New York, 1971

[BH71] Bott, R. R; Haefliger. A, On characteristic classes of $\Gamma$-foliations.Bull Bull. Amer. Math. Soc.78(1972), 1039–1044.

Recall that Bott's obstruction for integrability [Bott71] asserts that:

Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$.

In [p. 144, Hae71], Haefliger says:

($\ast$) The theorem of Bott ... implies that the induced homomorphism $$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$ is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r\geq 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.)

Can anyone explain why Bott's theorem implies ($\ast$)? I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations.

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

P.S. In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in bijection with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.

References

[Bott71] Bott. R, On topological obstructions to integrability. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 27–36 Gauthier-Villars Éditeur, Paris, 1971

[Hae71] Haefliger, A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163 Lecture Notes in Math., Vol. 197 Springer-Verlag, Berlin-New York, 1971

[BH71] Bott, R, On characteristic classes of $\Gamma$-foliations.Bull. Amer. Math. Soc.78(1972), 1039–1044.

Recall that Bott's obstruction for integrability [Bott71] asserts that:

Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$.

In [p. 144, Hae71], Haefliger says:

($\ast$) The theorem of Bott ... implies that the induced homomorphism $$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$ is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r\geq 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.)

Can anyone explain why Bott's theorem implies ($\ast$)? I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations.

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

P.S. In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in bijection with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.

References

[Bott71] Bott. R, On topological obstructions to integrability. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 27–36 Gauthier-Villars Éditeur, Paris, 1971

[Hae71] Haefliger. A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163 Lecture Notes in Math., Vol. 197 Springer-Verlag, Berlin-New York, 1971

[BH71] Bott. R; Haefliger. A, On characteristic classes of $\Gamma$-foliations. Bull. Amer. Math. Soc.78(1972), 1039–1044.

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Ken
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Ken
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