Homotopically rigid Sullivan algebras and Their applications
Cristina Costoya, David M\'endez, Antonio Viruel
TL;DR
This paper constructs homotopically rigid spaces and uses them to create highly connected rational spaces with specific symmetry properties, as well as inflexible and chiral manifolds, advancing understanding in rational homotopy theory.
Contribution
It introduces an infinite family of homotopically rigid spaces and applies them to build highly connected rational spaces and manifolds with prescribed symmetries.
Findings
Constructed an infinite family of homotopically rigid spaces
Developed highly connected rational spaces with finite self-homotopy groups
Produced highly connected inflexible and strongly chiral manifolds
Abstract
In this paper we construct an infinite family of homotopically rigid spaces. These examples are then used as building blocks to forge highly connected rational spaces with prescribed finite group of self-homotopy equivalences. They are also exploited to provide highly connected inflexible and strongly chiral manifolds.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Homotopically rigid Sullivan algebras and Their applications
Cristina Costoya
Departamento de Computación, Universidade da Coruña, Campus de Elviña, s/n, 15071 - A Coruña, Spain.
,
David Méndez
Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, E-29071-Málaga, Spain
and
Antonio Viruel
Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, E-29071-Málaga, Spain
IRMP, Université catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium
University of Regina, Department of Mathematics and Statistics
College West, Regina, CANADA
(Date: January 1, 1994 and, in revised form, June 22, 1994.)
Abstract.
In this paper we construct an infinite family of homotopically rigid spaces. These examples are then used as building blocks to forge highly connected rational spaces with prescribed finite group of self-homotopy equivalences. They are also exploited to provide highly connected inflexible and strongly chiral manifolds.
Key words and phrases:
Self-homotopy equivalences; Rational homotopy theory; Inflexible manifolds; Poincaré duality
1991 Mathematics Subject Classification:
Primary 55P10, 55P62; Secondary 57N65
The first author was partially supported by Ministerio de Economía y Competitividad (Spain), grant MTM2016-79661-P, and by Xunta de Galicia grant EM2013/16
The second author was partially supported by Ministerio de Educación, Cultura y Deporte grant FPU14/05137, by FEDER-MEC grants MTM2013-41768-P and MTM2016-78647-P, by Ministerio de Economía y Competitividad (Spain) grant MTM2016-79661-P, and by Xunta de Galicia grant EM2013/16
The third author was partially supported by FEDER-MEC grants MTM2013-41768-P and MTM2016-78647-P, and by Xunta de Galicia grant EM2013/16
\specialsection
*Introduction The group of self-homotopy equivalences of a space , , is rarely trivial. An obvious example is since . The first elaborated example with non trivial rational homology is constructed in [17] by D. Kahn. He expressed in [18] the belief that spaces with trivial group of self-homotopy equivalences, named homotopically rigid spaces, might play a role in some way of decomposing a space. Thus, the problem of determining for which is trivial becomes of interest, [18, Problem 3].
A decade after, Arkowitz and Lupton came across an example of a minimal Sullivan algebra (equivalently, a rational homotopy type of a space) with trivial, [3, Example 5.1]. This, let us say homotopically rigid algebra, provides us with motivation and guidance: in our paper [5], we use as a building block to construct minimal Sullivan algebras, [5, Definition 2.1], with the property that their group of self-homotopy equivalences are isomorphic to an arbitrary finite group previously fixed, [5, Theorem 1.1]. In other words, every finite group is realizable through minimal Sullivan algebras that are built upon ; this is a partial answer to Kahn’s realizability problem and we refer to [5] for more details. Observe that as an application, for the trivial group , is then a building block for an infinite family of homotopically rigid minimal Sullivan algebras (equivalently, homotopically rigid rational spaces). However, as interesting as it may seem, all the examples of the type above just mentioned are -connected since is -connected itself.
This paper came to light in our attempt to: (i) decide how rare homotopically rigid spaces are, since very few examples are known in literature. In that sense, we construct an infinite family of such spaces as highly connected as wanted; and (ii) show that building blocks other than Arkowitz-Lupton example can be chosen in the resolution of Kahn’s realizability problem in such a way that we are able to enlarge the family of examples in literature, all of them with low connectivity.
The paper is organized as follows. First, in Section 1 we construct an infinite family of highly connected minimal Sullivan algebras (see Definition 1.1) and we prove that they are homotopically rigid (see Theorem 1.6). Second, in Section 2 we show that algebras from Definition 1.1 serve as building blocks for new highly connected minimal Sullivan algebras (see Definition 2.1) with the property that their group of self-homotopy equivalences are isomorphic to any finite group that we fix (see Theorem 2.9 and Theorem 2.10). In other words, we are realizing the group through minimal Sullivan algebras that can be chosen as highly connected as desired. Section 3 consists of applications of the previous ones to differential geometry: we enlarge the class of inflexible manifolds existing in literature (see Theorem 3.1), and create new strongly chiral manifolds (see Corollary 3.3, and Corollary 3.4). We end this paper with an Appendix by Pascal Lambrechts and Don Stanley that allows us to guarantee the existence of highly connected inflexible manifolds.
Acknowledgments. The authors would like to thank the referee for carefully reading our manuscript and for giving such valuable comments which helped improving the quality of the paper.
1. Highly connected homotopically rigid CDGA
In this section an infinite family of highly connected homotopically rigid minimal Sullivan algebras are constructed. They extend the family of examples in literature, all of them of low connectivity.
We refer to [9] for basic facts in rational homotopy theory. Only simply connected -algebras of finite type are considered here. If is a graded rational vector space, we write for the free commutative graded algebra on . This is a symmetric algebra on tensored with an exterior algebra on . A Sullivan algebra is a nilpotent commutative differential graded algebra (CDGA) which is free as commutative graded algebra on a simply connected graded vector space of finite dimension in each degree. It is minimal if in addition . A Sullivan algebra is pure if on and . We recall that the geometric realization functor of Sullivan [21] gives the equivalence of categories between the homotopy category of minimal Sullivan algebras, and the homotopy category of rational simply connected spaces of finite type. Henceforward, by abuse of language, we sometimes refer to minimal Sullivan algebras as rational spaces.
Recall that the homotopy rigid algebra constructed by Arkowitz and Lupton [3, Example 5.1] is
[TABLE]
where dimensions and differential are
[TABLE]
Trying to obtain highly connected homotopy rigid algebras by re-scaling the degree of the generators in is hopeless: the differential in leads to a system of linear equations, involving the degree of the generators, whose unique solution is the one given by .
We now introduce the main building blocks of this paper.
Definition 1.1**.**
For any even integer, , we define the minimal Sullivan algebra
[TABLE]
where
[TABLE]
Remark 1.2**.**
Observe that since is even, only have even degree; the rest of generators have odd degree. This is a technicality which proves to be crucial in the proof of Theorem 1.6, the main result in this section.
For the sake of clarity, the proof of Theorem 1.6 is split into three lemmas. Recall that the algebras are introduced in Definition 1.1.
Lemmas on generators of
The first two lemmas measure how the degrees of the generators are, in some sense, far apart.
Lemma 1.3**.**
Generators of satisfy the following inequalities:
[TABLE]
Proof.
All the inequalities are straightforward except for the last one, which follows from the fact that is a factor in a term of , and has no generator in degree . ∎
Lemma 1.4**.**
None of the following integers are divisible by or by :
Proof.
We first show that they are not divisible by , which is an even number. Let us write . Then, for , Table 1 becomes:
Let be any of the polynomials in the table above. If is divisible by , then is so. Every monomial in is divisible by but the independent term, which is at most thus not divisible by . Therefore neither of and is divisible by .
In a similar way we prove that they are not divisible by , which is also an even number. Let us write . For Table 1 becomes:
Again, let be any of the polynomials in the table above. If is divisible by , then and are divisible by . Every monomial in is divisible by but the independent term, which is at most thus not divisible by . Therefore neither of and is divisible by , and is not divisible by . ∎
The third and final lemma describes some specific elements in . By we denote the elements of degree .
Lemma 1.5**.**
Let be such that . Then is a multiple of , for . Moreover, if is a cocycle, then it is also a coboundary.
Proof.
For , then which, by Lemma 1.4, is not divisible by or by . In this situation, has not monomials that are pure powers of or and, as a consequence, is a multiple of . Resulting from that for some with which, again by Lemma 1.4, is not divisible by or by . The same argument as above shows that is a multiple of , so for some . Then, .
Moreover, if is a cocycle, leading to . This equality forces and to be multiples of and respectively. Then, and for some and the equality above becomes Therefore and
[TABLE]
which exhibits as a coboundary. ∎
Main theorem in Section 1
We have all the ingredients to present and prove the main result in this section. Recall that , for any even , are the -connected algebras introduced in Definition 1.1. The following result directly implies that are homotopically rigid algebras:
Theorem 1.6**.**
The monoid of the homotopy classes of self-maps of has only two elements: the zero map and the identity.
Proof.
Let . By degree reasons, following Lemma 1.3 we know that and , , . We point out here that since is of odd degree, no summand from can be part of it.
While allow us to immediately obtain the following relations
[TABLE]
comparing is more demanding. On one hand we have that
[TABLE]
and on the other hand
[TABLE]
Note that none of the summands in (1.2) is a multiple of , whereas appears as a summand in (1.3). This forces . Also, if we compare monomials of the form , with , in (1.2) and (1.3), we are led to the conclusion that . This implies that is a cocycle and, by Lemma 1.5, a coboundary. Following the strategy of comparing the rest of the coefficients in both equations and considering (1.1) we get
[TABLE]
When , it is easy to obtain that . In other case, as and also , we get that , or equivalently, . Combining both identities, we have that . Therefore , and since , we deduce that . So, when , we infer that .
Summarizing all the steps, we have proved that for , there exists such that From there, it is only necessary to observe that is homotopically equivalent to and we have succeeded in proving that the monoid of the homotopy classes of self-maps of has only two elements: the zero map (for ), and the identity (for ). ∎
2. Realizing finite groups by highly connected CDGA
The purpose of this section is to prove that algebras from Definition 1.1 serve as building blocks in the resolution of Kahn’s realizability problem. Recall that Kahn’s problem asks, for an arbitrary group , if for some space ; if so, is said to be realized by . We argue as in [5, §2]: for any finite group , by Frucht’s theorem [11], there exists a finite, connected and simple graph whose automorphism group, , is isomorphic to . The landscape changes now from groups to graphs, and our approach consists on realizing the group by the following minimal Sullivan algebra:
Definition 2.1**.**
Let be an integer, and be a finite, connected and simple graph with more than one vertex. We define
[TABLE]
where
[TABLE]
Remark 2.2**.**
Observe that is described as a tensorial product where the left factor is the minimal Sullivan algebra from Definition 1.1, and the right factor codifies . Indeed, for every vertex , is represented by the generator , while codifies the neighborhood of , that is, the edges in that are incident to the vertex . Finally, notice that we are considering simple non oriented graphs, i.e. consists of non ordered pairs of vertices, and therefore the pairs and represent the same element in .
Lemmas on generators of
We prove now three technical lemmas that are needed for the main result in this section, Theorem 2.9.
Lemma 2.3**.**
Let us consider , and . The following integers are not divisible by either or :
Proof.
Nothing is to prove for the two top rows, it follows from Lemma 1.4 by taking . For the rest, we start by showing that they are not divisible by , which is an even number. Let us write , and . Then, the four bottom rows become:
Let be any of the polynomials in the table above: if is divisible by , then is so. Every monomial in is divisible by but the independent term, which is at most thus not divisible by . Therefore neither of nor is divisible by .
Similar arguments show that none of the integers from the table is divisible by , which is also an even number. Let us write , and . Then, the four bottom rows in Table 2 become:
Again, let be any of the polynomials in the table above: if is divisible by , then and are divisible by . Every monomial in is divisible by but the independent term, which is at most thus not divisible by . Hence neither of nor is divisible by . ∎
The description of some specific elements in the algebra is also needed:
Lemma 2.4**.**
Let and . If y_{i}A_{i}\in\big{(}{M}_{n}(\mathcal{G})\big{)}^{m}, for , , or , then , where . Moreover, if is a cocycle, then it is a coboundary.
Proof.
If y_{i}A_{i}\in\big{(}{M}_{n}(\mathcal{G})\big{)}^{m}, then . When , , or , is respectively the -th term of the first, third or fifth row in Table 2. By Lemma 2.3, as is not divisible by neither nor , we infer that is not a pure power of nor , so , where . Now, for , , or , is, respectively, the -th term of the second, fourth or sixth row in Table 2. Hence is not divisible by neither nor , and the same arguments allow us to write . Thus, we finally obtain that , where .
Similar arguments as in the proof of Lemma 1.5 show that if is a cocycle then it is a coboundary; details are omitted. ∎
The last technical result that we need is the following.
Lemma 2.5**.**
There is no pair of positive integers such that
[TABLE]
for . Moreover, there is no pair of non-negative integers verifying (2.1) for .
Proof.
To obtain the general solution of the diophantine equation (2.1), first notice that
[TABLE]
so the greatest common divisor of and is . Thus it suffices to compute a particular solution , and express the general one as
[TABLE]
For , the pair is a particular solution for (2.1) and the general solution is \big{(}k(18n+11),5n+3-k(15n+9)\big{)}, . It is clear that for , the second component is negative, and for , the first component is non positive. Thus there is no solution where both integers are positive.
For the rest of cases we just write the general solution below here. It is straightforward to check that at least one of the two components of the pair is negative for all :
∎
Sullivan algebras are elliptic
First, we recall that a Sullivan algebra is said to be elliptic when both and are finite-dimensional. The cohomology of an elliptic Sullivan algebra verifies Poincaré duality [14]. One can compute the degree of its fundamental class (a fundamental class of a Poincaré duality algebra , is a generator of , is said to be the formal dimension of the algebra) by the formula [9, Theorem 32.6]:
[TABLE]
where are the degrees of the elements of a basis of and of a basis of .
The existence of a fundamental class for elliptic Sullivan algebras gives rise to a notion of degree for self maps.
Definition 2.6**.**
Let be an elliptic Sullivan algebra with fundamental class . Given we say that the degree of is , , if .
The following result is inspired in classical works by Hopf [15, 16]:
Proposition 2.7**.**
Let be an elliptic Sullivan algebra and . Then is a self-homotopy equivalence if and only if .
Proof.
It is clear that a self homotopy equivalence has a non-zero degree since and the degree is a multiplicative invariant. Suppose now that . Then, we are going to prove that is injective. Indeed, assume that here exist a non trivial class such that . Now, since is a Poincaré duality algebra, consider the fundamental class and the (non-degenerate) bilinear form
[TABLE]
given by . This bilinear form induces an isomorphism
[TABLE]
given by . Therefore, since , there exists such that . Then which implies , contradicting our initial assumption. Hence is injective and, as we are considering finite type algebras, is a quasi-isomorphism, which means that is a self-homotopy equivalence. ∎
Finally, we prove the following lemma.
Lemma 2.8**.**
The algebra is an elliptic Sullivan algebra of formal dimension .
Proof.
We have to prove the cohomology of is finite-dimensional. Instead, we prove the equivalent condition that the pure Sullivan algebra associated to ,[9, Proposition 32.4], is also elliptic. This pure algebra, denoted by , has the same generators as and differentials
[TABLE]
The cohomology of is finite-dimensional because
[TABLE]
and the cohomology class
[TABLE]
Hence is elliptic and its formal dimension is obtained by (2.2). ∎
Main theorem in Section 2
We present the main result in this section. For the minimal Sullivan algebra from Definition 2.1, we prove:
Theorem 2.9**.**
For any finite simple graph , the -connected algebra realizes , i.e.,
[TABLE]
Before getting into the details of the proof, one immediate consequence:
Theorem 2.10**.**
Any finite group can be realized by infinitely many highly connected rational spaces.
Proof.
In [11], Frucht proves that every finite group is the group of automorphisms of a finite simple graph . It suffices then to apply Theorem 2.9. ∎
For the proof of Theorem 2.9, two more lemmas are needed. Recall from Remark 2.2 that algebras are constructed to codify the graph in terms of vertices (generators ) and their neighborhoods (generators ). Since an automorphism of is just a permutation of vertices that preserve neighborhoods, we deduce a first lemma. Explicitly:
Lemma 2.11**.**
Every induces defined by
[TABLE]
Moreover, given , then if and only if .
Proof.
Given , it is clear that . Finally, if , and , then for every the elements and must by equal up to decomposable elements, that is . Therefore for every , and . ∎
For the second lemma, we argue as in [5, Lemma 2.4]:
Lemma 2.12**.**
For every f\in\operatorname{End}\big{(}{M}_{n}(\mathcal{G})\big{)}, one of the following holds:
- (1)
* is an automorphism, and there exists such that*
[TABLE]
where is the morphism given by Lemma 2.11. 2. (2)
.
In particular, we deduce that for f\in\operatorname{End}\big{(}{M}_{n}(\mathcal{G})\big{)}, either or .
Proof.
We suppose that , that is, does not fall into case (2). We have to show that falls into case (1). The element f\in\operatorname{End}\big{(}{M}_{n}(\mathcal{G})\big{)} is uniquely determined by the image of generators of . First, we compute , , , , and . For degree reasons (see Lemma 1.3) we have:
[TABLE]
for . Since for , we get that
[TABLE]
Then, we continue with , for . Observe that no pure power of can be a summand in . Indeed , with , for every , which means that does not divide . Secondly, no summand of the form , with and both positive can appear in , since cannot be expressed as , with both positive, by Lemma 2.5. Then,
[TABLE]
where and , for all . Note that no term of the form or , for , can appear in the expression of (see Remark 1.2). Now, since , we get
[TABLE]
In this equation the sum of the terms having as a factor, respectively , , must all be zero. We start by paying attention to coefficients in . As
[TABLE]
we infer that , and therefore that must be a factor in and a factor in . Therefore for some , for some and , so that . We continue by paying attention to coefficients in . As
[TABLE]
then . But , so
[TABLE]
The same kind of arguments as above, imply now that and , for some .Then so . Finally, we pay attention to coefficients in . As
[TABLE]
we infer that . Since and ,
[TABLE]
so , for some . Then , and so .
Summarizing, we have , and , hence (2.5) can be rewritten as:
[TABLE]
Let us now prove that , with . Notice that if we succeed, as , the last summand in can be expressed as . It suffices to prove that and are not divisors of the degree of . This is clear as for , with , for and also for with which leads us to the conclusion that cannot be a pure power of either or . Then
[TABLE]
Third, we determine . Remark that terms of the form with cannot appear in the expression of , since by Lemma 2.5 there is no polynomial such that , for . It is then clear that
[TABLE]
where , , , for all , . Since and must coincide, on one hand using (2.3):
[TABLE]
and on the other hand:
[TABLE]
Now notice that on (2.8) there are no terms for , whereas in (2.9) there are. This leads to and thus to . Moreover, in (2.8) there are no terms at all involving , so we obtain , , and for all in (2.9). Hence d\big{(}\sum_{i=1}^{3}B_{i}(w)y_{i}\big{)}=0 and d\big{(}\sum_{i=1}^{3}C_{i}(u,w)y_{i}\big{)}=0, and by Lemma 2.4, we can finally reduce the expression to:
[TABLE]
Before computing the last image , , we work out here the value of the remaining constants. Compare again (2.8) to (2.9) and invoke (2.4) to obtain:
[TABLE]
what is a particular case of Equation (1.4) for . Therefore either or . Since , we know that (Proposition 2.7), and therefore , from which it follows that .
Finally, we end by computing , . Since , the degree reasons used to describe in (2.7) apply, and can be expressed as:
[TABLE]
with , and , for all , .
Then,
[TABLE]
On the other hand, using (2.6)
[TABLE]
First, notice that there is no summand in (2.12) of the form , where , and are pairwise distinc. This forces (2.13) to have at most two nontrivial coefficients . Furthermore, in (2.12) there is no factor of the form , where , which forces exactly one non trivial coefficient in (2.13) (recall that , hence at least one coefficient must be nontrivial). Consequently, in (2.13) there is a unique summand containing and, in (2.12) there can only be one non trivial coefficient . This fact can be read as there is a self-map of , , such that a\big{(}v,\sigma(v)\big{)} and c\big{(}v,\sigma(v)\big{)} are the only non trivial coefficients. Second, notice that in (2.13) there is no factor that is a pure power of , and thus, in (2.12). Also, since the graph has no loops, a term of type does not appear in (2.12), and a\big{(}v,\sigma(v)\big{)}\neq 0 implies that . We focus now on (2.13). Notice that
[TABLE]
and if two terms of this form are multiplied, their product is zero (). Thus,
[TABLE]
Observe that only one term containing appears in (2.14), which is multiplied by d\big{(}P(v)y_{1}y_{2}y_{3}\big{)}=P(v)(x_{1}^{3}x_{2}y_{2}y_{3}-x_{1}^{2}x_{2}^{2}y_{1}y_{3}+x_{1}x_{2}^{3}y_{1}y_{2}). However, in (2.12) there is no term containing both and . Therefore, we can conclude that , for every , and hence
[TABLE]
Now, comparing terms containing , and in (2.12) and (2.15), we get:
[TABLE]
In other words , and are cocycles in degrees , and respectively. We now invoke Lemma 2.4 to conclude that they are also coboundaries. Then, we can finally write
[TABLE]
for some m_{z_{v}}\in\big{(}M_{n}(\mathcal{G})\big{)}^{|z|-1}, and
[TABLE]
Comparing both (2.15) and (2.16), we get
[TABLE]
Therefore a\big{(}v,\sigma(v)\big{)}^{2}=a\big{(}w,\sigma(w)\big{)}, if . Observe that since is not a directed graph, is also an edge of , then a\big{(}w,\sigma(w)\big{)}^{2}=a\big{(}v,\sigma(v)\big{)} and a\big{(}v,\sigma(v)\big{)}^{4}=a\big{(}v,\sigma(v)\big{)}. Moreover, since is connected, and a\big{(}v,\sigma(v)\big{)}\neq 0, then a\big{(}v,\sigma(v)\big{)}=c\big{(}v,\sigma(v)\big{)}=1 for all . Therefore we can write:
[TABLE]
We shall see that in this case is, in fact, an element in . We first show that is a full endomorphism of the graph , that is, that if and only if \big{(}\sigma(v),\sigma(w)\big{)}\in E(\mathcal{G}). Indeed, if and only if there is a summand in , and this is equivalent to there being a summand in , or equivalently, \big{(}\sigma(v),\sigma(w)\big{)}\in E(\mathcal{G}). Now, since for every , , is one-to-one on the neighborhood of every vertex. Thus , which finally proves that if , then falls into case (1) of Lemma 2.12. ∎
Gathering Lemmas 2.11, and 2.12, we have succeeded in proving the following:
Lemma 2.13**.**
The monoid of homotopy classes of self-maps of is:
[TABLE]
As a consequence, we obtain our main result, Theorem 2.9.
Proof of Theorem 2.9: From Proposition 2.7, we know that self-maps with are not self-homotopy equivalences. Then, using Lemma 2.11, we conclude that , for every .
3. Inflexible manifolds
A closed, oriented and connected manifold is said to be inflexible if the set of all the possible degrees of its continuous self-maps is finite. As the degree is multiplicative, this condition is equivalent to ask for the set of all the possible degrees to be a subset of . Inflexible manifolds naturally appear within the framework of functorial seminorms on singular homology developed by Gromov [13, 12] and derived degree theorems (e.g. [6, Remark 2.6]): let be a closed, oriented and connected manifold with fundamental class . If there exists a functorial seminorm on singular homology such that , then is inflexible. In this way, oriented closed connected hyperbolic manifolds are shown to be inflexible; they do have non trivial simplicial volume, the value of -seminorm applied to the fundamental class [12, Section 0.3]. But -seminorm is trivial on simply connected manifolds [12, Section 3.1], which led Gromov to raise the question of whether every functorial seminorm on singular homology is trivial on all simply connected spaces [13, Remark (b) in 5.35]. This question is solved in the negative in [6] by constructing functorial seminorms associated to simply connected inflexible manifolds. Therefore, inflexible manifolds are extraordinary objects and still not many examples are known. Indeed, all the examples found in literature show low levels of connectivity when observing their minimal Sullivan models [3, 1, 6, 5].
In this section, we closely follow the lines of [5] and provide new examples of inflexible manifolds whose Sullivan models are as highly connected as desired (see Corollary 3.2). To that purpose, we first need to recall some tools.
First, we recall that for elliptic Sullivan algebras, the Barge and Sullivan obstruction theory ([4, 21]) decides if there exists a manifold (over ) of the same rational homotopy type of that algebra. Roughly speaking, the obstruction theory is trivial when the formal dimension of the algebra is not congruent to . Our examples will fall into this case.
Secondly, we recall a construction from [5, Proposition 3.1, Lemma 3.2]. Let be a -connected elliptic Sullivan algebra of formal dimension . Then, we can construct a -connected elliptic Sullivan algebra of formal dimension as follows: we choose a representative of the fundamental class of , let us say with and we define . The algebra inherits properties of : it is elliptic, the monoid of self-maps of coincides with the monoid of self-maps of , and finally, the connectivity is preserved.
At this point, we wish to explain our strategy. We have proved that from Definition 2.1 is an elliptic Sullivan algebra of even dimension, let us say , (see Lemma 2.8). By means of the construction above, we define a new elliptic Sullivan algebra of formal dimension . The monoid of self-maps of both algebras coincide, hence {\mathcal{E}}\big{(}\tilde{M}_{n}(\mathcal{G})\big{)} is finite which implies that is inflexible. Also and have the same connectivity. Now, as we have already mentioned, the obstruction of Barge and Sullivan is trivial and has the rational homotopy type of a manifold. Even more, by Proposition A.1, has the rational homotopy type of , a closed smooth manifold with the same connectivity as .
We can now prove the main theorem in this section.
Theorem 3.1**.**
For any finite group and any integer , is the group of self-homotopy equivalences of the rationalization of an inflexible manifold which is -connected.
Proof.
Take a graph such that [11]. For any , consider the algebra from Definition 2.1. We know from Theorem 2.9 that {\mathcal{E}}\big{(}M_{n}(\mathcal{G})\big{)}\cong\operatorname{Aut}(\mathcal{G}). We also know from Lemma 2.8 that is an elliptic minimal Sullivan algebra of formal dimension an even integer . We apply [5, Proposition 3.1] to obtain an elliptic Sullivan algebra of formal dimension , which we denote , such that {\mathcal{E}}\big{(}\tilde{M}_{n}(\mathcal{G})\big{)}\cong{\mathcal{E}}\big{(}M_{n}(\mathcal{G})\big{)}\cong G, by Lemma 2.13. Since is elliptic, every non zero degree map must be a self-homotopy equivalence by Proposition 2.7, and since {\mathcal{E}}\big{(}\tilde{M}_{n}(\mathcal{G})\big{)} is finite, the set of degrees is finite as well and is an inflexible Sullivan algebra. Now , so by Proposition A.1, is the rationalization of a -connected manifold . Observe that is also inflexible since so is . Thus
[TABLE]
where is the rational homotopy type of .
∎
In particular, we immediately obtain the following.
Corollary 3.2**.**
There exist infinitely many non-homotopically equivalent inflexible manifolds as highly connected as desired.
We end with some results on the existence of strongly chiral manifolds, that is, manifolds that do not admit orientation-reversing self-maps of degree (see [20], [1] or [5]).
Corollary 3.3**.**
For any and , there exists a rationally -connected strongly chiral manifold of dimension .
Proof.
Let be a connected simple graph with vertices, and let be the corresponding manifold obtained in the proof of Theorem 3.1 above, its minimal Sullivan algebra being . Any self-map of is shown in [5, Lemma 3.2] to verify that , thus it has degree [math] or . ∎
Manifolds provided by Corollary 3.3 can be used to construct inflexible and strongly chiral product manifolds by exploiting techniques from [19]. In [19, Example 3.7], it is shown that given a simply connected inflexible manifold one can construct an inflexible product manifold just by considering a closed oriented inflexible non-simply connected manifold that does not admit maps of non-zero degree from direct products and . Here we prove an “inverse”:
Corollary 3.4**.**
Let be a non necessarily simply connected closed oriented inflexible (resp. strongly chiral) manifold that does not admit maps of non-zero degree from direct products. Then there exist simply connected strongly chiral manifolds such that and is inflexible (resp. strongly chiral).
Proof.
Let , and be an integer such that . Let be any of the -connected strongly chiral manifolds from Corollary 3.3. Since , any continuous map maps to [math]. Thus the pair is under the assumptions of [19, Theorem 1.4], and the result follows from [19, Corollary 1.5(c)] (resp. [19, Corollary 1.5(a)]). ∎
Appendix A By Pascal Lambrechts and Don Stanley
Realization of rationally Poincaré duality complexes by highly connected bounding manifolds
Proposition A.1**.**
Let be a -connected rational space () whose rational cohomology algebra satisfies Poincaré duality in dimension . When is a multiple of , suppose furthermore that the quadratic form on has signature [math] and descends from a quadratic form over . Then there exists a -connected closed smooth manifold rationally equivalent to and that is a boundary of a compact oriented manifold.
Proof.
Let be some trivial vector bundle of rank over . Then the Thom space of , , is homotopy equivalent to which is a wedge of rational spheres. Since has Poincaré duality in dimension there is a homotopy class whose image under the Hurewicz map is non trivial. One can pick a representative of transverse to and then the preimage is a smooth closed -manifold with trivial stable normal bundle which comes with a map non trivial in . Since is -connected, the classical surgery argument makes -connected. Further surgeries of dimensions can turn into a rational homology equivalence up to the middle dimension. As explained in [21, Section 13] or [4] the obstruction to do surgery in the middle dimension in our case vanish if the dimension is not a multiple of or if the dimension is a multiple of and the quadratic form on is equivalent to (that is it descends from a quadratic form over and the signature vanishes). This gives a -connected manifold with a rational homotopy equivalence . Furthermore all the Pontrjagin and Stiefel-Whitney numbers of vanish because it has a trivial stable normal bundle, which implies that , and hence , is the boundary of an oriented compact manifold. ∎
Remark A.2**.**
Jim Fowler and Zhixu Su have proved that we cannot drop the hypothesis that the signature vanishes when the dimension is a multiple of , even if is realizable by a smooth manifold. Indeed, consider the unique simply-connected rational homotopy type with with . This rational homotopy type is realizable by a closed smooth manifold but it cannot be -connected ([10, Theorem B]).
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] M. Amann, Mapping degrees of self-maps of simply-connected manifolds , Int. Math. Res. Notices 18 (2015), 8545–8589.
- 2[2] M. Arkowitz, Problems on self-homotopy equivalences, in: Groups of homotopy self-equivalences and related topics , Contemp. Math. 274 (2001), 309–315.
- 3[3] M. Arkowitz, G. Lupton, Rational obstruction theory and rational homotopy sets , Math. Z. 235 (2000), no. 3, 525–539.
- 4[4] J. Barge, Structures différentiables sur les types d’homotopie rationnelle simplement connexes , Ann. Sci. École Norm. Sup. (4), 9 (1976), no. 4, 469–501.
- 5[5] C. Costoya, A. Viruel, Every finite group is the group of self homotopy equivalences of an elliptic space , Acta Mathematica 213 (2014), 49–62.
- 6[6] D. Crowley, C. Löh, Functorial seminorms on singular homology and (in)flexible manifolds , Algebraic & Geometric Topology 15 (2015), 1453–1499.
- 7[7] Y. Félix, La dichotomie elliptique-hyperbolique en homotopie rationnelle. Astérisque, 176 (1989).
- 8[8] Y. Félix, S. Halperin, J.C. Thomas, Elliptic spaces , Bull. Amer. Math. Soc. 25 (1991), 69–73.