nLab Gauss-Manin connection

Contents

Contents

Definition

A Gauss-Manin connection is a canonical flat connection on fiber bundles of relative ordinary (but possibly twisted) (co-)homology groups, hence a way of transforming (co-)homology classes as a parameter varies.

Details

(…)

For fiber bundles

The existence of a flat connection on bundles of fiberwise-cohomology groups is easy to understand in the case that the fibers form a locally trivial fiber bundle: In this case the fiberwise cohomology of the local trivialization of this fiber bundle is a local trivialization of the bundle of its fiberwise cohomology groups. The Gauss-Manin connection in this case is made explicit for instance in Voisin 2002, above Def. 9.13.

Simple as this may appear, at least in the case of fiberwise twisted cohomology this case subsumes (for forgetful fibrations of configuration spaces of points in the plane) the profound example of solutions to the Knizhnik-Zamolodchikov equation, identified as such via the hypergeometric integral construction – this is highlighted as such in Etingof, Frenkel & Kirillov 1998, §7.5, reproduced as Example below.

In the following we give (following SS22) a more abstract/formal (model category theoretic/homotopy theoretic) argument for the existence and construction of the Gauss-Manin connection for twisted cohomology on a class of fiber bundles which subsume the case that gives the KZ-equation, namely projections of configuration spaces of points (Ex. below).

This following more abstract argument has the advantage that it applies at once to all generalized cohomology-theories for which classifying spaces exist (Whitehead-generalized cohomology theories and more general non-abelian cohomology-theories, such as unstable Cohomotopy), and that its structure is readily formulated in general ( , 1 ) (\infty,1) -toposes and in homotopy type theory. In fact, in that formal language the following construction of the Gauss-Manin connection for twisted cohomology on fibers of a fiber bundle is so simple that it becomes essentially a tautology.

For non-abelian cohomology

The following is a (non-standard) general-abstract argument for the existence and construction of the structure of local systems on fiberwise cohomology groups, which applies in the generality of Whitehead-generalized cohomology theories, and in fact of non-abelian cohomology theories.

The simple idea – using that all these notions of cohomology have classifying spaces – is that the fiberwise 0-truncation of the fiberwise mapping space to that classifying space provides at once the covering space which exhibits the local system.


We consider the following situation, see also (6) and (14).

Let

  1. BkHauskTopB \in kHaus \hookrightarrow kTop be

    1. a compactly generated Hausdorff space,

    2. locally path-connected and semi-locally simply connected,

  2. (Xp XB)kTop /B(X \xrightarrow{p_X} B) \in kTop_{/B} be

    1. a compactly generated topological space (i.e. a k-space),

    2. in the slice category over BB, via a Hurewicz fibration

      (1)p XhFib. p_X \,\in\, hFib \,.
    3. which is locally trivial over a numerable cover {U jι jB} jJ\big\{ U_j \xhookrightarrow{\;\iota_j\;} B\big\}_{j \in J} for some typical fiber X 0X_0:

      (2)jJι j *(X,p X)p B *X 0 \underset{j \in J}{\forall}\;\;\;\; \iota_j^\ast (X, p_X) \,\simeq\, p_B^\ast X_0
    4. what has the structure of a CW-complex:

      (3)X 0CWCplkTop. X_0 \,\in\, CWCpl \to kTop \,.

For bBb \in B we write X bX_b for the fiber of p Xp_X over bb:

(4)X b X (pb) p X {b} B \array{ X_b &\xrightarrow{\phantom{----}}& X \\ \big\downarrow &{}_{{}^{(pb)}}& \big\downarrow{}^{\mathrlap{p_X}} \\ \{b\} &\underset{\phantom{----}}{\hookrightarrow}& B }

Example

(forgetful map between configuration spaces of points)
For n,Nn,N \,\in\, \mathbb{N}, the map of configuration spaces of ordered points in the plane

X=Conf{1,,n+N}( 2)Conf{1,,N}( 2)=B X \;=\; \underset{ \{ 1, \cdots, n+N \} }{Conf} \big( \mathbb{R}^2 \big) \xrightarrow{\;\;\;} \underset{ \{1, \cdots, N\} }{Conf} \big( \mathbb{R}^2 \big) \;=\; B \,

(which forgets the position of the first nn among n+Nn + N points) satisfies the above assumption, by this Prop. and this Prop..

Definition

(non-abelian cohomology sets)
For any AkTopA \,\in\, kTop, and for X bX_b a CW-complex, we write

(5)A 0(X b)H 0(X b;A)π 0Map(X b,A) A^0(X_b) \;\coloneqq\; H^0(X_b;\, A) \;\coloneqq\; \pi_0 Map \big( X_b ,\, A \big)

for the non-abelian cohomology of X bX_b with coefficients in AA.

Example

(ordinary complex cohomology)
For A=K(n,)A \,=\, K(n,\mathbb{C}) an Eilenberg-MacLane space (here of the additive abelian group of complex numbers), Def. yields ordinary complex cohomology

H n(X b;)π 0Map(X b,K(,n)). H^n(X_b; \mathbb{C}) \,\simeq\, \pi_0 Map\big(X_b, \, K(\mathbb{C},n)\big) \,.

Example

(Cohomotopy)
For A=S nA \,=\, S^n the n n -sphere, Def. yields Cohomotopy

π n(X b)π 0Map(X b,S n). \pi^n(X_b) \,\simeq\, \pi_0 Map\big(X_b, \, S^n\big) \,.

Proposition

(fiberwise non-abelian cohomology sets form local system)
For any k-space AkTopA \,\in\, kTop, the non-abelian cohomology-sets (5) of the fibers X bX_b (4) with coefficients in AA constitute a local system over BB, in that they arrange into a covering space over BB (equivalently, they are the values of a functor from the fundamental groupoid of BB to Sets).

(6)

Moreover, if {U jι jB} jJ\big\{U_j \xhookrightarrow{\iota_j} B\big\}_{j \in J} is an open cover over which XBX \to B locally trivializes with typical fiber X 0X_0, then the covering space of cohomology sets locally trivializes over the same cover, with typical fiber H 0(X 0;A)H^0(X_0; A).

Proof

Write p B *AkTop /Bp_B^\ast A \,\in\, kTop_{/B} for the base change of AA to the slice over BB, i.e. for the trivial fibration

(7)p B *AB×Apr BB. p_B^\ast A \;\;\;\coloneqq\;\;\; B \times A \xrightarrow{\; pr_B \;} B \,.

This is clearly a Hurewicz fibration. Since also p Xp_X is such by assumption (1), it follows (by this Prop.) that the fiberwise mapping space between the two (this Def.) is also a Hurewicz fibration, hence its fibers are homotopy fibers and as such coincide with the ordinary mapping space between the fibers (by this Example):

(8)Map(X b,A) Map((X,p X),p B *A) (pb) Fib {b} B. \array{ Map \big( X_b ,\, A \big) & \xrightarrow{\phantom{--}} \;\;\;\;\; & \phantom{---} \mathclap{ Map \big( (X,p_X) ,\, p_B^\ast A \big) } \phantom{---} \\ \big\downarrow &{}_{{}^{(pb)}}& \big\downarrow{}^{ \mathrlap{\in Fib}} \\ \{b\} &\xrightarrow{\phantom{-----}}& B \mathrlap{\,.} }

Hence we may think of this diagram equivalently as exhibiting the homotopy pullback (in the classical model structure on topological spaces) of the mapping fibration along {b}B\{b\} \to B.

But since homotopy pullback preserves fiberwise 0-truncation (by this Prop.) it follows that the fiberwise 0-truncation of the fiberwise mapping space fibration has as homotopy fibers the non-abelian cohomology-sets (5):

H 0(X b;A)=π 0Map(X b,A) π 0/BMap((X,p X),p B *A) (pb) Fib {b} B. \array{ \mathllap{ H^0(X_b;\, A) \;=\; } \pi_0 Map \big( X_b ,\, A \big) & \xrightarrow{\phantom{--}} \;\;\;\;\;\;\; & \phantom{----} \mathclap{ \pi_{0/B} \, Map \big( (X,p_X) ,\, p_B^\ast A \big) } \phantom{----} \\ \big\downarrow &{}_{{}^{(pb)}}& \big\downarrow{}^{ \mathrlap{\in Fib} } \\ \{b\} &\xrightarrow{\phantom{-----}}& B \mathrlap{\,.} }

Here by fibrant replacement we may assume that the right vertical map is again a Serre fibration, as indicated, in which case it is a covering space-projection whose fibers are the desired cohomology groups. This proves the first statement.

The second statement follows by the same argument, after using over any patch U jι jBU_j \xhookrightarrow{\iota_j} B of the given open cover the following natural identification:

ι j *Map((X,p X),p B *A) Map(ι j *(X,p X),ι j *p B *A) by (9) Map(p U j *X 0,p U j *A) by (2) p U j *Map(X 0,A) by (9) \begin{array}{ll} \iota_j^\ast Map \big( (X, p_X) ,\, p_B^\ast A \big) \\ \;\simeq\; Map \big( \iota_j^\ast (X, p_X) ,\, \iota_j^\ast p_B^\ast A \big) & \text{by (9)} \\ \;\simeq\; Map \big( p_{U_j}^\ast X_0 ,\, p_{U_j}^\ast A \big) & \text{by (2)} \\ \;\simeq\; p_{U_j}^\ast Map \big( X_0 ,\, A \big) & \text{by (9)} \end{array}

Here we have used that pullback is a closed functor with respect to fiberwise mapping spaces (by this Prop.):

(9)f *Map((X,p X),(Y,p Y))Map(f *(X,p X),f *(Y,p Y)). f^\ast Map \big( (X,p_X) ,\, (Y,p_Y) \big) \;\simeq\; Map\big( f^\ast(X,p_X),\, f^\ast(Y,p_Y) \big) \,.

For twisted non-abelian cohomology

We generalize the above discussion to twisted non-abelian cohomology (Def. below).

The argument is essentially the same as in the previous Prop. , only that now:

  1. the base spaces starts out being the product space B×BGB \times B G of the previous base space BB with the classifying space BGB G for the twists,

  2. before passing to homotopy classes of fiberwise maps we form the right base change (B×p BG) *(B \times p_{B G})_\ast (dependent product) along this classifying space.

To see that the previous argument generalizes to this case one needs to

  1. observe that this right base change is right Quillen (Lem. below),

  2. use a Beck-Chevalley relation (15) to see that it is compatible with pullback along {b}B\{b\} \to B.


For GG a discrete group, we write BG|N(G*)|B G \,\coloneqq\, \vert N (G \rightrightarrows \ast)\vert for the topological realization of the nerve of the delooping groupoid of GG, hence for the usual classifying space (which, by discreteness of GG, is an Eilenberg-MacLane space K(G,1)K(G,1)) in its usual realization as a CW-complex:

(10)BGCWCpxkTop cof. B G \,\in\, CWCpx \hookrightarrow kTop^{cof} \,.

For AGAct(kTop)A \,\in\, G Act(kTop) a topological G G -space (a space equipped with a continuous map GG-action), its Borel construction is – regarded as the AA-fiber bundle associated to the universal G G -principal bundle – an object in the slice category over the classifying space BGB G (10)

(11)(A× GEG,p A× GEG)kTop /BG \big( A \times_G E G ,\, p_{A \times_G E G } \big) \;\; \in \; kTop_{/B G}

Lemma

For BkTopB \in kTop, the right base change along the projection pr B:B×BGBpr_B \,\colon\, B \times B G \to B

kTop B×BG Qu(pr B) *(pr B) *kTop B kTop_{B \times B G} \underoverset { \underset{ (pr_B)^\ast }{\longrightarrow} } { \overset{ (pr_B)^\ast }{\longleftarrow} } {\bot_{{}_{\mathrlap{Qu}}}} kTop_{B}

is a Quillen adjunction between the respective slice model categories of the classical model structure on topological spaces.

Proof

Since pr B:B×BGBpr_B \,\colon\, B \times B G \to B is just projection out of a Cartesian product, the pullback (pr B) *(pr_B)^\ast acts by forming the product topological space with the space BGB G. But since BGB G is cofibrant (10) this operation preserves cofibrations and acyclic cofibrations of kTop QukTop_{Qu} (by this Prop.) and hence also those of its slice model structure.

Definition

(twisted non-abelian cohomology, as in FSS20, §2.2)
For

the τ b\tau_b-twisted non-abelian cohomology of X bX_b with coefficients in AA is the connected components of the space of sections of the pullback bundle of A× GEGA \times_G E G along τ b\tau_b:

(12)A τ(X b)H τ(X b;A) π 0Γ X b(τ b *(A× GEG)) π 0((p BG) *Map((X b,τ b),(A× GEG,p A× GEG))), \begin{aligned} A^\tau(X_b) \;\coloneqq\; H^\tau(X_b;\, A) & \;\coloneqq\; \pi_0 \, \Gamma_{X_b} \big( \tau_b^\ast (A \times_G E G) \big) \\ & \;\simeq\; \pi_0 \bigg( (p_{B G})_\ast Map \Big( (X_b, \tau_b) ,\, \big( A \times_G E G ,\, p_{A \times_G E G} \big) \Big) \bigg) \,, \end{aligned}

where the codomain on the right is the Borel construction (11) on AA.

Remark

To see that (p BG) *Map((,),(,))(p_{B G})_\ast Map\big( (-,-),\, (-,-) \big) in the last line of (12) is really the same as the space of sections, use the Yoneda lemma (for kTop opkTop^{op}) on the following sequence of natural isomorphisms:

kTop(U,(p BG) *Map((X b,τ b),(A× GEG,p A× GEG))) kTop /BG((p BG) *U,Map((X b,τ b),(A× GEG,p A× GEG))) kTop /BG(((p BG) *U)×(X b,τ b),(A× GEG,p A× GEG)) kTop /BG((τ b) !(U×X b,pr X b),(A× GEG,p A× GEG)) by (13) kTop /X b((U×X b,pr X b),(τ b) *(A× GEG,p A× GEG)) kTop(U×X b,(τ b) *(A× GEG))×kTop(U×X b,X b){pr X b} kTop(U,Map(X b,(τ b) *(A× GEG))×Map(X b,X b){id X b}) kTop(U,Γ X((τ b) *(A× GEG))) \begin{array}{ll} kTop \Big( U ,\, (p_{B G})_\ast Map \big( (X_b, \tau_b) ,\, (A \times_G E G, p_{A \times_G E G}) \big) \Big) \\ \;\simeq\; kTop_{/ B G} \Big( (p_{B G})^\ast U ,\, Map \big( (X_b, \tau_b) ,\, (A \times_G E G, p_{A \times_G E G}) \big) \Big) \\ \;\simeq\; kTop_{/ B G} \Big( \big( (p_{B G})^\ast U \big) \times (X_b, \tau_b) ,\, (A \times_G E G, p_{A \times_G E G}) \Big) \\ \;\simeq\; kTop_{/ B G} \Big( (\tau_b)_! (U \times X_b, pr_{X_b}) ,\, (A \times_G E G, p_{A \times_G E G}) \Big) & \text{by (13)} \\ \;\simeq\; kTop_{/ X_b} \Big( (U \times X_b, pr_{X_b}) ,\, (\tau_b)^\ast (A \times_G E G, p_{A \times_G E G}) \Big) \\ \;\simeq\; kTop \big( U \times X_b ,\, (\tau_b)^\ast (A \times_G E G) \big) \underset{ kTop \big( U \times X_b ,\, X_b \big) }{\times} \big\{ pr_{X_b} \big\} \\ \;\simeq\; kTop \Big( U ,\, Map \big( X_b ,\, (\tau_b)^\ast (A \times_G E G) \big) \underset{ Map \big( X_b ,\, X_b \big) }{\times} \big\{ \mathrm{id}_{X_b} \big\} \Big) \\ \;\simeq\; kTop \Big( U ,\, \Gamma_X \big( (\tau_b)^\ast ( A \times_G E G ) \big) \Big) \end{array}

Here each step is the hom-isomorphism of various adjoint functors, except the marked one which is immediate from this pullback square:

(13)U×X b pr X b X b (pb) τ b U×BG pr BG BG. \array{ U \times X_b &\overset{pr_{X_b}}{\longrightarrow}& X_b \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow\mathrlap{{}^{\tau_b}} \\ U \times B G &\underset{pr_{B G}}{\longrightarrow}& B G \mathrlap{\,.} }

and remembering that left base change (τ b) !(\tau_b)_! is given simply by post-composition with τ b\tau_b.

Now we may state and prove the twisted-cohomology generalization of Prop. :

Proposition

(fiberwise twisted non-abelian cohomology sets form local system)
With (X,p X)(X, p_X) a fiber bundle as above, and given a global twist on its total space

τ:XBG, \tau \,\colon\, X \xrightarrow{\;\;} B G \,,

the τ b\tau_b-twisted AA-cohomology sets (Def. ) of X bX_b, for any bBb \in B, arrange into a covering space over BB, generalizing the un-twisted statement from Prop. .

(14)

Proof

In the evident generalization of (8), we have the following homotopy pullback diagram:

Map(((X b,τ b)),(A× GEG,p A× GEG)) Map((X,(p X,τ)),p B *(A× GEG,p A× GEG)) (pb) Fib {b}×BG B×BG. \array{ \phantom{---} \mathclap{ Map \Big( \big( (X_b, \tau_b) \big) ,\, \big( A \times_G E G, p_{A \times_G E G} \big) \Big) } \phantom{-------} & \!\!\!\!\!\!\!\!\!\!\!\! \xrightarrow{\phantom{---}} \;\;\;\;\;\;\;\;\;\; & \phantom{-----} \mathclap{ Map \Big( \big( X, (p_X, \tau) \big) ,\, p_B^\ast \big( A \times_G E G, p_{A \times_G E G} \big) \Big) } \phantom{-----} \\ \big\downarrow &{}_{{}^{(pb)}}& \big\downarrow{}^{ \mathrlap{\in Fib}} \\ \{b\} \times B G &\xrightarrow{\phantom{-----}}& B \times B G \mathrlap{\,.} }

Now by the Beck-Chevalley relation

(15)(b ()) *(B×p BG) *(p BG) *(b ()×BG) * \big( b_{(-)} \big)^\ast \circ \big( B \times p_{B G} \big)_\ast \;\; \simeq \;\; \big( p_{B G} \big)_\ast \circ \big( b_{(-)} \times B G \big)^\ast

this gives the following pullback diagram:

(p BG) *Map(((X b,τ b)),(A× GEG,p A× GEG)) (B×p BG) *Map((X,(p X,τ)),p B *(A× GEG,p A× GEG)) (pb) Fib {b} B, \array{ \phantom{---} \mathclap{ (p_{B G})_\ast Map \Big( \big( (X_b, \tau_b) \big) ,\, \big( A \times_G E G, p_{A \times_G E G} \big) \Big) } \phantom{---------} & \!\!\!\!\!\!\!\!\!\!\!\! \xrightarrow{\phantom{---}} \;\;\;\;\;\;\;\;\;\; & \phantom{--------} \mathclap{ (B \times p_{B G})_\ast Map \Big( \big( X, (p_X, \tau) \big) ,\, p_B^\ast \big( A \times_G E G, p_{A \times_G E G} \big) \Big) } \phantom{-----} \\ \big\downarrow &{}_{{}^{(pb)}}& \big\downarrow{}^{ \mathrlap{\in Fib}} \\ \{b\} &\xrightarrow{\phantom{----------}}& B \mathrlap{\,,} }

where the vertical map on the right is still a fibration, by Lem. .

From here we conclude as in the proof of Prop. that for all bBb \in B we have:

H τ b(X b;A)=π 0(p BG) *Map(((X b,τ b)),(A× GEG,p A× GEG)) π 0/B(B×p BG) *Map((X,(p X,τ)),p B *(A× GEG,p A× GEG)) (pb) Fib {b} B, \array{ \phantom{---} \mathclap{ \mathllap{ H^{\tau_b}(X_b;\, A) \,=\, } \pi_0 (p_{B G})_\ast Map \Big( \big( (X_b, \tau_b) \big) ,\, \big( A \times_G E G, p_{A \times_G E G} \big) \Big) } \phantom{---------} & \!\!\!\!\!\!\!\!\!\!\!\! \xrightarrow{\phantom{---}} \;\;\;\;\;\;\;\;\;\; & \phantom{---------} \mathclap{ \pi_{0/B} (B \times p_{B G})_\ast Map \Big( \big( X, (p_X, \tau) \big) ,\, p_B^\ast \big( A \times_G E G, p_{A \times_G E G} \big) \Big) } \phantom{-----} \\ \big\downarrow &{}_{{}^{(pb)}}& \big\downarrow{}^{ \mathrlap{\in Fib}} \\ \{b\} &\xrightarrow{\phantom{----------}}& B \mathrlap{\,,} }

so that the fibration on the right is the covering space which exhibits the claimed local system.

Example

(Hypergeometric Knizhnik-Zamologchikov equations)
Let

  1. p X:XBp_X \colon X \to B be the forgetful map of configuration spaces of points from Ex.

    Conf n+N( 2)Conf N( 2), Conf_{n + N}(\mathbb{R}^2) \xrightarrow{\;\;} Conf_{N}(\mathbb{R}^2) \,,
  2. A=K(,n)A \,=\, K(\mathbb{C},n) the Eilenberg-MacLane space classifying ordinary complex cohomology;

  3. G/kU(1)G \,\coloneqq\, \mathbb{Z}/k \,\subset\, U(1) be a cyclic group, acting by multiplication by roots of unity on the complex numbers \mathbb{C} and hence on K(,n)K(\mathbb{C},n).

Then Prop. constructs the Gauss-Manin connection claimed in Etingof, Frenkel & Kirillov 1998, §7.5.

Examples

References

Original articles:

Textbook accounts:

and with focus on the special case of surjective submersions of smooth manifolds:

Lecture notes:

Gauss-Manin connections over configuration spaces of points:

and review in the context of hypergeometric solutions to the Knizhnik-Zamolodchikov equation:

The discussion above follows:

See also:

Discussion in cyclic homology:

  • Boris Tsygan, On the Gauss-Manin connection in cyclic homology, Methods Funct. Anal. Topology 13 (2007), no. 1, 8394.

  • Ezra Getzler, Cartan homotopy formulas and the Gauss/Manin connection in cyclic homology, pdf

In noncommutative geometry:

Last revised on June 27, 2022 at 11:29:37. See the history of this page for a list of all contributions to it.