A note on Gorenstein spaces
Yves Felix, Steve Halperin

TL;DR
This paper explores the properties of Gorenstein spaces through homotopy invariants associated with differential graded algebras, establishing conditions under which certain cohomology algebras exhibit Poincaré duality.
Contribution
It introduces a homotopy invariant for augmented differential graded algebras and demonstrates its behavior under Sullivan extensions, linking algebraic properties to topological duality.
Findings
${ m dim}\, { m extbf{T}}(R)=1$ iff $H(R)$ is Poincaré duality algebra.
${ m extbf{T}}( ext{W} ext{⊗} ext{Z})= { m extbf{T}}( ext{W}) ext{⊗} { m extbf{T}}( ext{Z})$ for Sullivan extensions with finite-dimensional cohomology.
Finite-dimensional Poincaré duality properties transfer from universal covers to base spaces under certain conditions.
Abstract
Associated with an augmented differential graded algebra is a homotopy invariant . This is a graded vector space, and if is the ground field and then dim if and only if is a Poincar\'e duality algebra. In the case of Sullivan extensions in which dim we show that This is applied to finite dimensional CW complexes where the fundamental group acts nilpotently in the cohomology of the universal covering space. If is a Poincar\'e duality algebra and and are finite dimensional then they are also Poincar\'e duality algebras.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
A note on Gorenstein spaces
Yves Félix and Steve Halperin
Abstract
Associated with an augmented differential graded algebra is a homotopy invariant . This is a graded vector space, and if is the ground field and then dim if and only if is a Poincaré duality algebra. In the case of Sullivan extensions in which dim we show that
[TABLE]
This is applied to finite dimensional CW complexes where the fundamental group acts nilpotently in the cohomology of the universal covering space. If is a Poincaré duality algebra and and are finite dimensional then they are also Poincaré duality algebras.
Keywords: Poincaré duality algebras, rational homotopy theory, Gorenstein algebras.
In this note we work over an arbitrary ground field . A Poincaré duality algebra is then a graded algebra such that and the pairing
[TABLE]
defines an isomorphism , . In particular, is necessarily finite dimensional.
A Poincaré complex at is then a CW complex whose cohomology is a Poincaré duality algebra. In this note we develop the properties of path connected spaces satisfying a more general homotopy condition, which reduces to Poincaré duality when the cohomology is finite dimensional. As it turns out, this provides additional flexibility which enables applications to fibrations.
This ”Gorenstein” condition is defined via the generalization (cf. the Appendix) by Eilenberg and Moore of the classical Ext and Tor functors to the category of modules over a differential graded algebra (dga). More precisely, given a based path connected space we denote by and the singular chain and cochain complexes for , and by and their respective homology. In particular, the Alexander-Whitney diagonal makes into an augmented dga, and we denote
[TABLE]
Definition. is Gorenstein at if dim . In this case for some , and is called the Gorenstein degree of .
Note that the cap product makes into a right -module and that the identification
[TABLE]
is an isomorphism of -modules. It follows that is the dual,
[TABLE]
and so is Gorenstein at if and only if
[TABLE]
The connection with Poincaré duality is then provided by
Theorem 1. Suppose an augmented path connected space satisfies , some . Then the following conditions are equivalent:
- (i)
is Gorenstein at . 2. (ii)
dim Tor. 3. (iii)
is a Poincaré duality algebra.
proof: (i) (iii) By hypothesis, , and so has a minimal -semi free resolution . In particular,
[TABLE]
since the differentials in vanish. Since dim Tor it follows that is the free -module on a single generator . Let be the cycle . Then and so the cap product induces an isomorphism of -modules
[TABLE]
Thus is a Poincaré duality algebra.
(iii) (ii). If is a Poincaré duality algebra then the cap product makes into a free -module with a single generator, the fundamental class. This gives (ii).
(ii) (i). this is immediate if Tor is computed via an Eilenberg-Moore semi free resolution for (cf. the Appendix).
The definitions above extend in the obvious way to the category of augmented dga’s : we set
[TABLE]
and is Gorenstein if dim. In particular, a quasi-isomorphism of augmented dga’s induces an isomorphism via the isomorphisms
[TABLE]
For simplicity in this category we adopt the notation:
[TABLE]
and
[TABLE]
when is a dga and is an -module.
This permits the application of dga homotopy theory to the study of Gorenstein spaces in general and Poincaré complexes in particular. The relevant definitions and Lemmas from dga homotopy theory are collected in the Appendix at the end of this note. In particular we introduce the invariants
[TABLE]
for an augmented dga and a path connected based space . Then we have
Proposition 1. If is a graded vector space of finite type, then
[TABLE]
In particular, is Gorenstein if and only if dim
proof: It follows from the hypotheses that the natural inclusion is a quasi-isomorphism of -modules, and so
[TABLE]
In this setting the analogue of Theorem 1 is
Theorem 2. Suppose is a cdga. If and then the following conditions are equivalent
- (i)
dim 2. (ii)
dim 3. (iii)
is a Poincaré duality algebra.
In this case is Gorenstein.
proof. The same argument as in the proof of Theorem 1 shows that there is a non-degree preserving isomorphism . Therefore dim and is Gorenstein. Moreover, as in the proof of Theorem 1, as this is an isomorphism of -modules, it follows that is a Poincaré duality algebra. This in turn implies that dim and so dim. Finally using an Eilenberg-Moore semifree resolution for gives that (ii) (i).
For the rest of this Introduction we restrict to the case .
If is any based, path connected space then is connected by quasi-isomorphisms to the minimal Sullivan model for ([3]). This permits the application of Sullivan’s theory to establish
Theorem 3. Suppose is a fibration of based path connected spaces in which dim, and acts nilpotently in . Then
[TABLE]
If, additionally, either has finite type, or else has finite type and , then
[TABLE]
In this case is Gorenstein at if and only if is a Poincaré duality algebra and is Gorenstein at .
Corollary. Suppose is a finite dimensional CW complex with fundamental group and universal covering space . If acts nilpotently in each and if is a Poincaré duality algebra, then each of the conditions dim, has finite type, has finite type, is equivalent to the condition : is a Poincaré duality algebra. When these hold, is Gorenstein at .
proof. First observe that any of these three conditions implies that dim. In fact, because is a finite dimensional complex it follows that for some , and so, if dim, Theorem 2 asserts that dim. Of course if has finite type then dim. Finally, by hypothesis, dim and so if has finite type, and have Sullivan models of at most countable dimension. Let
[TABLE]
be the -extension determined by the fibration . Thus and , are at most countably dimensional, so this also holds for . But by Theorem 5.1 in [4], ; Therefore each dim, and so dim.
Now suppose dim. Then Theorem 3 applies and gives
[TABLE]
Since is a Poincaré duality algebra, dim (Theorem 2). Thus dim, and so is a Poincaré duality algebra. Moreover, since and dim it follows that is a graded vector space of finite type. Now Lemma 3.2 in [4] asserts that has finite type, and the first three conditions of the Corollary hold. Since then has finite type Proposition 1 implies that is Gorenstein at .
The hypothesis on in the Corollary is essential, and does not always hold even when and are closed manifolds, as the following Example shows.
Example. . In this example, and . The universal cover, then may be described as follows: Consider first the connected sum of with . The union of this space with will be denoted by . We then consider a family of spaces homeomorphic to and indexed by the integers. In each we denote by the sphere and by the sphere . Then
[TABLE]
As a CW complex has the rational homotopy type of . The action of is the identity on and is the translation in the wedge .
The model of the projection is
[TABLE]
with , , . A model for the fiber of the model is given by with . It follows that a basis for the homology of the fiber of the model is given by and , .
By a result of Dwyer [1] for a fibration with nilpotent base, the homology of the fiber of the model is the subalgebra of in which acts nilpotently. Here , and with trivial action. The space is the space of series . The invariant elements are the linear combinations
[TABLE]
Theorem 3 is a rational homotopy theory analogue of a theorem of Gottlieb ([6]) which states that in a fibration of connected spaces which have the homotopy type of finite CW complexes, then is a Poincaré complex if and only if and are. In [2] the authors introduce the application of dga homotopy theory to establish Theorem 3 under additional finiteness assumptions, and with the hypothesis that the spaces are simply connected. With these additional hypotheses Murillo [7] extends the result of [2] to ground fields of arbitrary characteristic.
1 Fibrations and Sullivan models
In this section and all cdga’s satisfy , and we recall the necessary elements from the theory of Sullivan models.
A Sullivan algebra is a cdga of the form , where the underlying algebra is the free commutative graded algebra on . The differential is required to satisfy the Sullivan condition: where is an increasing filtration and .
More generally, a -extension is a cdga morphism , , in which , and is an increasing filtration satisfying . In particular, is a semifree -module and an augmentation of extends to the augmentation, , of given by . If , this is called a Sullivan extension.
If then has a -extension with . This is denoted and is called an acyclic closure for . In particular, an augmentation is an -semifree resolution, and thus
[TABLE]
On the other hand, with each path connected space are associated its Sullivan models; these are Sullivan algebras and are connected by dga quasi-isomorphisms to . Thus if is a Sullivan model for then
[TABLE]
Moreover, with a fibration
[TABLE]
of path connected spaces is associated a -extension
[TABLE]
in which is a Sullivan model for , is a Sullivan model for , and .
Proposition 2. With the hypotheses and notation above, suppose in the fibration that dim and acts nilpotently in . Then is a Sullivan model for , and so
[TABLE]
Moreover, if the additional hypotheses of Theorem 3 are satisfied, then each of , , and are graded vector spaces of finite type. In this case,
[TABLE]
proof. Theorem 5.1 in [4] asserts that is a Sullivan model for . If the additional hypotheses are satisfied, either has finite type or else has finite type and . In the first case Proposition 3.8 in [4] asserts that has finite type. In the second case, because and dim, it follows that has finite type. Thus Lemma 3.2 in [4] asserts that has finite type. The final assertion then follows from Proposition 1.
2 Proof of Theorem 3
Fix a -extension
[TABLE]
in which and . Denote by , and let , and respectively be acyclic closures of , and .
Theorem 4. Suppose in the -extension above that dim. Then
[TABLE]
In particular, if has finite type then is Gorenstein if and only if and are.
Before embarking on the proof of Theorem 4, consider the special case that and are graded vector spaces of finite type. In view of Proposition 2 in the previous section, applied when is a Sullivan model for , Theorem 3 is then an immediate consequence of Theorem 4.
For the proof of Theorem 4, we recall that
[TABLE]
The proof is in multiple steps, and we recall from the Appendix that denotes the underlying graded algebra of a dga . .
Proposition 3. With the hypotheses of Theorem 4,
- (i)
The -modules is a homotopically semi-free -module. 2. (ii)
Composition defines a morphism
[TABLE]
of -modules which is also an -homotopy equivalence.
proof: Decompose as the direct sum
[TABLE]
where the differential satisfies and . Thus and so dim. Then define a morphism
[TABLE]
of -modules by setting
[TABLE]
and
[TABLE]
Filtering by the degree in shows that is a quasi-isomorphism. It follows that is the direct sum of the -modules ( denotes the underlying graded algebra of a dga )
[TABLE]
in which . Division by the first two defines a differential in and a surjective quasi-isomorphism
[TABLE]
of -modules. Note that restricts to the retraction determined by (3).
Now, because is a -extension there is an increasing filtration , , such that , , and . This filtration projects under to a filtration with the corresponding properties with respect to . In particular, is -semifree. Since is a quasi-isomorphism of -semifree modules, it is an -homotopy equivalence. This implies in turn that
[TABLE]
is also an -homotopy equivalence.
Now since dim, in the filtration some . Filter by the submodules defined by
[TABLE]
A simple calculation shows that , , and that
[TABLE]
It follows that is -semifree and so is a homotopy semifree -module. This establishes (i).
To prove (ii) observe first that is by definition a morphism of -modules. Moreover, it follows from (i) that
[TABLE]
is an -homotopy equivalence. On the other hand, since dim
[TABLE]
and so
[TABLE]
Thus we have the commutative diagram
[TABLE]
in which and are -homotopy equivalences. If follows that is an -homotopy equivalence.
Corollary.
[TABLE]
is a quasi-isomorphism of -modules.
proof: This follows because is -semi-free and is a quasi-isomorphism of -modules.
Next note that the inclusion makes into an -module.
Proposition 4. There is an -flat quasi-isomorphism
[TABLE]
proof: Set and observe that
[TABLE]
On the other hand, the inclusion makes into a semifree -module. In fact we may write the -module in the form
[TABLE]
where . Then set
[TABLE]
so that each is a -module and
[TABLE]
Then we may write
[TABLE]
Since is a homotopically semi-free -module and is -semi-free it follows that is homotopically -semi-free. In particular, it is -flat. For simplicity write
[TABLE]
Now observe that each
[TABLE]
is an -flat module. It follows by induction on that each is -flat and hence is -flat. Moreover, since is a quasi-isomorphism of -modules it then follows that
[TABLE]
are surjective quasi-isomorphisms.
The next step is to construct maps of complexes
[TABLE]
such that , extends and . Suppose is constructed. Since is a surjective quasi-isomorphism it has a right inverse . Then
[TABLE]
Because , there is a map such that
[TABLE]
Extend to a map
[TABLE]
and set . Then set . Since each is a quasi-isomorphism so are each and .
Further, and extend uniquely to morphisms of -modules
[TABLE]
and
[TABLE]
The commutative diagram
[TABLE]
shows that each is a quasi-isomorphism. Thus so is .
We now show by induction that each is -flat. In fact, and induce a quotient quasi-isomorphism
[TABLE]
which may be identified as a quasi-isomorphism
[TABLE]
from a semi-free -module to a homotopically semi-free -module. Such a morphism is a homotopy equivalence and hence is -flat. Thus if is -flat so is , and thus so is .
But
[TABLE]
and so the Proposition is proved.
Recall that if is a sub dga and then we write
[TABLE]
In particular, if is any -module then the projection
[TABLE]
induces an -module structure in . Similarly, if is a -module then inherits a -module structure. But
[TABLE]
and so is an -module.
Proposition 5. If is a semi-free -module and is an -module then
[TABLE]
is an isomorphism.
proof: An exact sequence and direct limit argument reduces the Proposition to the case . In this case has the form
[TABLE]
But
[TABLE]
which identifies as the identity.
proof of Theorem 4: As above we denote and .
Since is -semi-free, Proposition 3 provides a quasi-isomorphism
[TABLE]
Write the left hand side as
[TABLE]
Proposition 4 then provides an -flat quasi-isomorphism
[TABLE]
From this we obtain a quasi-isomorphism
[TABLE]
It remains to show that
[TABLE]
Since is semi-free, Proposition 5 provides a quasi-isomorphism
[TABLE]
Moreover, there is a natural inclusion of -modules
[TABLE]
which we show is an -homotopy equivalence. In fact, in the proof of Proposition 3 we constructed a homotopy equivalence of -modules,
[TABLE]
in which dim. This yields the commutative diagram
[TABLE]
in which the vertical arrows are homotopy equivalences of -modules. But the bottom horizontal arrow is an isomorphism because dim. it follows that
[TABLE]
is an -homotopy equivalence. Applying therefore gives a quasi-isomorphism
[TABLE]
of -modules.
Now, is -semi-free, and it follows that
[TABLE]
But . Thus these quasi-isomorphisms combine to give
[TABLE]
But and , which establishes the isomorphism of the Theorem.
Finally, as observed at the start of this section, if has finite type so does . In this case , and are respectively the duals of , and This gives the last assertion of the Theorem. .
3 Sullivan extensions
In this section .
As recalled in Section 1 a fibration of path connected spaces determines a Sullivan extension
[TABLE]
in which is a Sullivan model for and is a Sullivan model for . Here and may be chosen to be minimal models: and . Moreover, if acts nilpotently in each and each dim then is a Sullivan model for .
Now consider the special case of the fibration described in the Corollary to Theorem 3, in which is a finite dimensional CW complex. Assume further that and are Poincaré duality algebras, and dim.
Theorem 5. Suppose in the minimal Sullivan model of that has finite type. Then, with the hypotheses above, and are Poincaré duality algebras.
Remark. If is the minimal Sullivan model of a path connected space then the sub dga is the Sullivan analogue of , and indeed the minimal Sullivan model of has the form . But it may well happen that .
proof ot Theorem 5. The acyclic closure of has the form with concentrated in degree [math]. Since dim it follows that
[TABLE]
satisfies for some . Since is assumed to have finite type this implies that dim.
Now by Theorem 4,
[TABLE]
But since is is a Poincaré duality algebra, this implies that and are 1-dimensional. In particular, is a Poincaré duality algebra. Finally, since dim, Proposition 9.6 in [4] asserts that for some . Now Theorem 2 asserts that is a Poincaré duality algebra.
4 An extension of Theorem 3
Here again, and we consider a fibration
[TABLE]
of path connected spaces with corresponding Sullivan extension
[TABLE]
Then and , but if does not act nilpotently in it may happen that and are not isomorphic.
In this setting we denote by the subalgebra of elements on which acts nilpotently, and we recall that for any -module , denotes the cohomology of with local coefficients in .
Theorem 6. Suppose with the notation above that has finite type and that dim. If
[TABLE]
then
[TABLE]
In particular, if is Gorenstein at then is a Poincaré duality algebra.
proof: Theorems 1 and 3 in [5] imply that , and so Theorem 6 follows from Theorem 4.
5 Appendix: Differential algebra
We establish the following conventions in the category of modules over differential graded algebras (dga) defined over an arbitrary ground field . The differentials are usually denoted by , but are normally suppressed from the notation: if is a differential graded object then denotes the underlying graded object. If is a commutative dga (cdga) then left -modules determine the corresponding right -modules via
[TABLE]
These are identified when is a commutative dga (cdga).
The functors and are simply denoted and . If and is an -module then is sometimes denoted by .
If is a dga morphism and is an R-module then is assigned the A- and R- module structures given by
[TABLE]
Thus this R-module structure coincides with that obtained from the A-module structure via precomposition with .
Two A-modules and have the same homotopy type if there are morphisms and such that both composites are homtopic to the respective identities via -linear maps.
Next recall that an A-module P is A-semifree if it can be equipped with an increasing filtration by A-modules such that and there are A-module isomorphisms
[TABLE]
In this case we have an isomorphism in which and .
A semifree resolution of an -module is a quasi-isomorphism from a semi free -module. These always exist and are unique up to quasi-isomorphism ([4, Chap.6]). Given a semi free resolution of an -module, filtering by the determines a spectral sequence whose -term has the form
[TABLE]
The semi free resolution is called an Eilenberg-Moore resolution if this sequence is a resolution of the -module . Every -module has an Eilenberg-Moore resolution ([3, Prop 20.11]).
The Eilenberg-Moore generalizations of Ext and Tor to dga-modules are defined by
[TABLE]
where is any semi free resolution of . In particular, if is a second -module then
[TABLE]
Now consider the special case of a dga in which and . Let be a direct summand of and set
[TABLE]
This defines a sub dga for which the inclusion is a quasi-isomorphism. Since converts semi-free -modules to semi free -modules the standard construction for -modules gives
Lemma A.1. Suppose a dga satisfies and . If an -module satisfies , some , then has an -semifree resolution such that .
Definition. A semi free -module is called minimal if .
Lemma A.2. Suppose is a morphism of -modules in which is augmented to , , and . If and admit minimal semi free resolutions, then the following conditions are equivalent:
- (i)
is a quasi-isomorphism 2. (ii)
is an isomorphism.
proof: That (i) (ii) is standard (see eg. [4, Chap.6]). To see that (ii) (i) replace by the dga described above to reduce to the case . Then lift up to homotopy to a morphism between minimal semi free resolutions for and . Since the differentials in and vanish, induces an isomorphism .
Now write and , and filter by the subspaces . Then the associated graded map induced by is an isomorphism, and so itself is an isomorphism and is a quasi-isomorphism.
More generally, we say an -module is homotopy semifree if it has the homotopy type of a semifree A-module.
Lemma A.3
- (i)
Suppose is a homotopy semifree A-module. Then preserves quasi-isomorphisms. 2. (ii)
A quasi-isomorphisms between semi-free -modules is a homotopy equivalence. 3. (iii)
If is any -module and if the A-modules M and N have the same homotopy type then so do the A-modules and . 4. (iv)
A quasi-isomorphism from a semi-free -module to a homotopy semifree -module is a homotopy equivalence.
proof: Immediate from the definitions.
Next, suppose is a cdga and . Then ([4, Chap.3]) there is a cdga morphism
[TABLE]
such that is injective, is -semifree, and . The cdga is called an acyclic closure for .
If is a cdga then an -module is flat if preserves quasi-isomorphisms. Also a quasi-isomorphism of -modules is flat if for any -module, , is a quasi-isomorphism.
Lemma A.4
- (i)
If is an exact sequence of -modules and and are flat, then so is . 2. (ii)
The direct limit of flat -modules is flat. 3. (iii)
A homotopically semifree -module is flat.
proof: (i) and (ii) are immediate. By [3] semifree -modules are flat, and it follows that so are homotopy semifree -modules.
Lemma A.5
- (i)
The direct limit of flat quasi-isomorphisms of -modules is a flat quasi-isomorphism. 2. (ii)
Suppose
[TABLE]
is a row exact commutative diagram of morphisms of -modules. If and are -flat so is .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] William Dwyer, Exotic convergence of the Eilenberg-Moore spectral sequence , Illinois J. of Math. 19 (1975), 607-617.
- 2[2] Yves Félix, Steve Halperin and Jean-Claude Thomas, On Gorenstein spaces , Advances in Mathematics 71 (1988), 92-112.
- 3[3] Yves Félix, Steve Halperin and Jean-Claude Thomas, Rational Homotopy Theory , Graduate Texts in Mathematics 215, Springer Verlag, 2000
- 4[4] Yves Félix, Steve Halperin and Jean-Claude Thomas, Rational Homotopy Theory II , World Scientific, 2015.
- 5[5] Yves Félix and Jean-Claude Thomas, Le Tor différentiel d’une fibration non nilpotente , Journal of Pure and Applied Algebra, 38 (1985), 217-233.
- 6[6] Daniel Gottlieb, Poincaré duality and fibrations , Proc. Amer. Math. Soc. 76 (1979), 148-150
- 7[7] Aniceto Murillo, The virtual Spivak fiber, duality on fibrations and Gorenstein spaces , Trans. Amer. math. Soc. 359 (2007), 3577-3587.
