nLab Gauss-Manin connection

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 (pdf), see also Etingof, Frenkel & Kirillov 1998, §7.5 (pdf).

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 $(\infty,1)$-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. $B \in kHaus \hookrightarrow kTop$ be

   1. a compactly generated Hausdorff space,

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

2. $(X \xrightarrow{p_X} B) \in kTop_{/B}$ be

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

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

      (1)$$p_X \,\in\, hFib \,.$$

   3. which is locally trivial over a numerable cover $\big\{ U_j \xhookrightarrow{\;\iota_j\;} B\big\}_{j \in J}$ for some typical fiber $X_0$:

      (2)$$\underset{j \in J}{\forall}\;\;\;\; \iota_j^\ast (X, p_X) \,\simeq\, p_B^\ast X_0$$

   4. what has the structure of a CW-complex:

      (3)$$X_0 \,\in\, CWCpl \to kTop \,.$$

For $b \in B$ we write $X_b$ for the fiber of $p_X$ over $b$:

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

Example

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

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

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

Definition

(non-abelian cohomology sets)
For any $A \,\in\, kTop$, and for $X_b$ a CW-complex, we write

(5)$$A^0(X_b) \;\coloneqq\; H^0(X_b;\, A) \;\coloneqq\; \pi_0 Map \big( X_b ,\, A \big)$$

for the non-abelian cohomology of $X_b$ with coefficients in $A$.

Example

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

$$H^n(X_b; \mathbb{C}) \,\simeq\, \pi_0 Map\big(X_b, \, K(\mathbb{C},n)\big) \,.$$

Example

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

$$\pi^n(X_b) \,\simeq\, \pi_0 Map\big(X_b, \, S^n\big) \,.$$

Proposition

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

(6)

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

Proof

Write $p_B^\ast A \,\in\, kTop_{/B}$ for the base change of $A$ to the slice over $B$, i.e. for the trivial fibration

(7)$$p_B^\ast A \;\;\;\coloneqq\;\;\; B \times A \xrightarrow{\; pr_B \;} B \,.$$

This is clearly a Hurewicz fibration. Since also $p_X$ 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)$$\array{ Map \big( X_b ,\, A \big) & \xrightarrow{\phantom{--}} \;\;\;\;\; & \phantom{---} \mathclap{ Map \big( (X,p_X) ,\, p_B^\ast A \big) } \phantom{---} \\ \big\downarrow &{}_{{}^{(pb)}}& \big\downarrow{}^{ \mathrlap{\in Fib}} \\ \{b\} &\xrightarrow{\phantom{-----}}& B \mathrlap{\,.} }$$

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 $\{b\} \to B$.

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):

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

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 $U_j \xhookrightarrow{\iota_j} B$ of the given open cover the following natural identification:

$$\begin{array}{ll} \iota_j^\ast Map \big( (X, p_X) ,\, p_B^\ast A \big) \\ \;\simeq\; Map \big( \iota_j^\ast (X, p_X) ,\, \iota_j^\ast p_B^\ast A \big) & \text{by (9)} \\ \;\simeq\; Map \big( p_{U_j}^\ast X_0 ,\, p_{U_j}^\ast A \big) & \text{by (2)} \\ \;\simeq\; p_{U_j}^\ast Map \big( X_0 ,\, A \big) & \text{by (9)} \end{array}$$

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

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

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 $B \times B G$ of the previous base space $B$ with the classifying space $B G$ for the twists,

2. before passing to homotopy classes of fiberwise maps we form the right base change $(B \times p_{B G})_\ast$ (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 $\{b\} \to B$.

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

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

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

(11)$$\big( A \times_G E G ,\, p_{A \times_G E G } \big) \;\; \in \; kTop_{/B G}$$

Lemma

For $B \in kTop$, the right base change along the projection $pr_B \,\colon\, B \times B G \to B$

$$kTop_{B \times B G} \underoverset { \underset{ (pr_B)^\ast }{\longrightarrow} } { \overset{ (pr_B)^\ast }{\longleftarrow} } {\bot_{{}_{\mathrlap{Qu}}}} kTop_{B}$$

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

Proof

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

Definition

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

• $G$ a discrete group,

• $A$ a $G$-space,

• $\tau_b \,\colon \,X_b \xrightarrow{\;\;} B G$ a map to the classifying space (10),

the $\tau_b$-twisted non-abelian cohomology of $X_b$ with coefficients in $A$ is the connected components of the space of sections of the pullback bundle of $A \times_G E G$ along $\tau_b$:

(12)$$\begin{aligned} A^\tau(X_b) \;\coloneqq\; H^\tau(X_b;\, A) & \;\coloneqq\; \pi_0 \, \Gamma_{X_b} \big( \tau_b^\ast (A \times_G E G) \big) \\ & \;\simeq\; \pi_0 \bigg( (p_{B G})_\ast Map \Big( (X_b, \tau_b) ,\, \big( A \times_G E G ,\, p_{A \times_G E G} \big) \Big) \bigg) \,, \end{aligned}$$

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

Remark

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

$$\begin{array}{ll} kTop \Big( U ,\, (p_{B G})_\ast Map \big( (X_b, \tau_b) ,\, (A \times_G E G, p_{A \times_G E G}) \big) \Big) \\ \;\simeq\; kTop_{/ B G} \Big( (p_{B G})^\ast U ,\, Map \big( (X_b, \tau_b) ,\, (A \times_G E G, p_{A \times_G E G}) \big) \Big) \\ \;\simeq\; kTop_{/ B G} \Big( \big( (p_{B G})^\ast U \big) \times (X_b, \tau_b) ,\, (A \times_G E G, p_{A \times_G E G}) \Big) \\ \;\simeq\; kTop_{/ B G} \Big( (\tau_b)_! (U \times X_b, pr_{X_b}) ,\, (A \times_G E G, p_{A \times_G E G}) \Big) & \text{by (13)} \\ \;\simeq\; kTop_{/ X_b} \Big( (U \times X_b, pr_{X_b}) ,\, (\tau_b)^\ast (A \times_G E G, p_{A \times_G E G}) \Big) \\ \;\simeq\; kTop \big( U \times X_b ,\, (\tau_b)^\ast (A \times_G E G) \big) \underset{ kTop \big( U \times X_b ,\, X_b \big) }{\times} \big\{ pr_{X_b} \big\} \\ \;\simeq\; kTop \Big( U ,\, Map \big( X_b ,\, (\tau_b)^\ast (A \times_G E G) \big) \underset{ Map \big( X_b ,\, X_b \big) }{\times} \big\{ \mathrm{id}_{X_b} \big\} \Big) \\ \;\simeq\; kTop \Big( U ,\, \Gamma_X \big( (\tau_b)^\ast ( A \times_G E G ) \big) \Big) \end{array}$$

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

(13)$$\array{ U \times X_b &\overset{pr_{X_b}}{\longrightarrow}& X_b \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow\mathrlap{{}^{\tau_b}} \\ U \times B G &\underset{pr_{B G}}{\longrightarrow}& B G \mathrlap{\,.} }$$

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

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

Proposition

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

$$\tau \,\colon\, X \xrightarrow{\;\;} B G \,,$$

the $\tau_b$-twisted $A$-cohomology sets (Def. ) of $X_b$, for any $b \in B$, arrange into a covering space over $B$, generalizing the un-twisted statement from Prop. .

(14)

Proof

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

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

Now by the Beck-Chevalley relation

(15)$$\big( b_{(-)} \big)^\ast \circ \big( B \times p_{B G} \big)_\ast \;\; \simeq \;\; \big( p_{B G} \big)_\ast \circ \big( b_{(-)} \times B G \big)^\ast$$

this gives the following pullback diagram:

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

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 $b \in B$ we have:

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

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

Example

(Hypergeometric Knizhnik-Zamolodchikov equations)
Let

1. $p_X \colon X \to B$ be the forgetful map of configuration spaces of points from Ex.

   $$Conf_{n + N}(\mathbb{R}^2) \xrightarrow{\;\;} Conf_{N}(\mathbb{R}^2) \,,$$

2. $A \,=\, K(\mathbb{C},n)$ the Eilenberg-MacLane space classifying ordinary complex cohomology;

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

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

Examples

• Picard-Fuchs equation

• Knizhnik-Zamolodchikov connection

Related concepts

• Ehresmann theorem

• flat connection

• flat vector bundle

• local system

References

Original articles:

• Yuri Manin, Algebraic curves over fields with differentiation, Izv. Akad. Nauk SSSR Ser. Mat. 22 6 (1958) 737-756 $[$mathnet:izv3998, pdf$]$ (in Russian)

• Nicholas M. Katz, On the differential equations satisfied by period matrices, Publications Mathématiques de l’IHÉS, Tome 35 (1968) 71-106 $[$numdamPMIHES_1968__35__71_0$]$

• Alexander Grothendieck, On the de Rham cohomology of algebraic varieties, Publications Mathématiques de l’IHÉS, Tome 29 (1966) 95-103 $[$numdam:PMIHES_1966__29__95_0$]$

• Sec. (3.2), (3.3), (3.4) in: A. Grothendieck, Crystals and the de Rham cohomology of schemes, in “Dix exposes sur la cohomologie des schemas”, Adv. Stud. Pure Math. 3, North Holland 1968, 306–358 pdf

• Nicholas M. Katz, Tadao Oda, On the differentiation of De Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8 2 (1968) 199-213 $[$doi:10.1215/kjm/1250524135$]$

• Egbert Brieskorn, Die Monodromie der isolierten Singularitäten von Hyperflächen, Masucripta Math. 2 (1970) 103-161 $[$doi:10.1007/BF01155695, pdf$]$

• Pierre Deligne, Equations différentielles à points singuliers réguliers, Lecture Notes Math. 163, Springer (1970) $[$publications.ias:355$]$

• Phillip A. Griffiths, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems, Bull. Amer. Math. Soc. 76 (1970) 228-296 $[$doi:10.1090/S0002-9904-1970-12444-2$]$

Textbook accounts:

• Pierre Deligne, Equations différentielles à points singuliers réguliers, Lecture Notes Math. 163, Springer (1970) $[$publications.ias:355$]$

• Valentine S. Kulikov, Part I of: Mixed Hodge Structures and Singularities, Cambridge University Press (1998) $[$doi:10.1017/CBO9780511758928$]$

• Alexandru Dimca, Theorem 2.5.14 in: Sheaves in Topology, Universitext, Springer (2004) $[$doi:10.1007/978-3-642-18868-8$]$

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

• Claire Voisin (translated by Leila Schneps), Section 9.2.1 (Def. 9.13) of: Hodge theory and Complex algebraic geometry I, Cambridge Stud. in Adv. Math. 76 (2002) [doi:10.1017/CBO9780511615344, def: pdf]

Lecture notes:

• Spencer Bloch, Gauss-Manin connections (2016) [pdf, pdf]

Gauss-Manin connections over configuration spaces of points:

• Daniel C. Cohen, Peter Orlik, Gauss-Manin connections for arrangements, I Eigenvalues, Compositio Math. 136 (2003) 299-316 $[$arXiv:math/0105063, doi:10.1023/A:1023262022279$]$

• Daniel C. Cohen, Peter Orlik, Gauss-Manin connections for arrangements, II Nonresonant weights, Amer. J. Math. 127 (2005) 569-594 $[$arXiv:math/0207114, jstor:40067930$]$

• Daniel C. Cohen, Peter Orlik, Gauss-Manin connections for arrangements, III Formal connections, Trans. Amer. Math. Soc. 357 (2005) 3031-3050 $[$arXiv:math/0307210, doi:10.1090/S0002-9947-04-03621-9$]$

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

• Pavel Etingof, Igor Frenkel, Alexander Kirillov, Section 7.5 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998) [ISBN:978-1-4704-1285-2, review pdf, def: pdf]

The discussion above follows:

• Gauss-Manin- and Knizhnik-Zamolodchikov connections via abstract homotopy theory, Chapter 4 of: David J. Myers, Hisham Sati, Urs Schreiber: Topological Quantum Gates in Homotopy Type Theory, Comm. Math. Phys. (2024) [arXiv:2303.02382]

See also:

• eom, Gauss-Manin connection

• Wikipedia, Gauss-Manin connection

• Allan Yashinski, Smooth deformations and the Gauss-Manin connection (arxiv/1410.0715)

Discussion in cyclic homology:

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

• Victor Ginzburg, Travis Schedler, Boris Tsygan, Free products, cyclic homology, and the Gauss–Manin connection, Advances in Mathematics 231:3-4 (2012) 2352–2389 doi

• Ezra Getzler, Cartan homotopy formulas and the Gauss/Manin connection in cyclic homology, in: “Quantum deformations of algebras and their representations,” Israel Math. Conf. Proc. 7 (1993) 65–78 pdf

In noncommutative geometry:

• Vasily Dolgushev, Dmitry Tamarkin, Boris Tsygan, Noncommutative calculus and the Gauss-Manin connection (arxiv/0902.2202)