Algebraic Topology of Special Lagrangian Manifolds

Algebraic Topology of Special Lagrangian Manifolds

Mustafa Kalafat    Eyüp Yalçınkaya
Abstract

In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of manifolds with calibrated geometries. We collect these results here for the sake of completeness. As applications of our algebraic topological study we present some results on special Lagrangian-free embeddings of surfaces and 3-manifolds into the Euclidean 4 and 6-space.

Keywords: Calibrations; special holonomy; fiber bundles; Grassmannians.

Mathematics Subject Classification 2010: Primary 53C38; Secondary 57R20, 57R22.

1 Introduction

This paper is devoted to the algebraic topological and geometric study of the space of oriented 3-planes in 6-dimensional real Euclidean space which we denote by . These manifolds are traditionally named as Grassmannians. We will give some definitions in the subject first, interested reader would like to consult to the fundamental article [harveylawson] of Harvey and Lawson or [joyceSLclay] for more background. Let be a Grassmannian manifold defined by all oriented -dimensional subspaces of For any there are orthonormal vectors such that Let be a closed -form on . If for any orthonormal set of vectors , then is called a calibration on The set

is called the contact set or face of the calibration . A k-submanifold is called calibrated if its tangent subplanes are in the contact set. An example of a calibration on is the real n-form The calibrated submanifolds in this geometry are Lagrangian submanifolds of which satisfy an additional ’determinant’ condition. They are therefore called special Lagrangian submanifolds. They, of course, have the property of being absolutely area-minimizing.

Let be a calibrated manifold. A -plane is said to be tangential to a submanifold if for some . A closed submanifold is called -free if there are no -planes which are tangential to . Each submanifold of dimension strictly less than the degree of is automatically -free. Locally, generic -dimensional submanifolds are -free. Depending on the calibration, there is an upper bound for the dimension of a -free submanifold. The free dimension of a calibrated manifold is the maximum dimension of a linear subspaces in which contains no -planes. Subspaces which satisfy such condition are called -free. Hence, the dimension of a -free submanifold can not exceed . For all well-known calibrations on manifolds with special holonomy, this dimension is computed and shown in the Table 1 of free dimensions. See [HLpotentialtheory] for the details.

Calibration Explanation Dimension Free dimension
Kähler and related 2p-forms 2n
Re() = Holomorphic Volume n-form 2n
Quaternionic 4p-calibrations 4n
associative 3-form 7 4
co-associative 4-form 7 4
Cayley 4-form 8 4
Table 1: Free dimensions.

-free submanifolds are the generalization of totally real submanifolds in complex geometry to calibrated manifolds. They are used to construct Stein-like domains in calibrated manifolds, called as -convex domains.

In our paper we investigate the invariants of this important Grassmannian. In particular we compute its integral homology as a main result of our paper.

Theorem 4.2. The homology of the oriented Grassmann manifold is given by

We also compute its cohomology ring.

Theorem LABEL:G3R6ring. The cohomology ring of the Grassmannian is as follows where .

Along the way we also compute the rings of some of the Stiefel manifolds.

Corollary 2.4. The cohomology ring of the Stiefel manifolds are the following for which .

In addition we compute some homotopy groups as well.

Lemma 3.1. The preliminary homotopy groups of the Grassmannian are the following.

Using Serre’s spectral sequence we also compute the invariant for the special Lagrangian manifold SLAG, which is defined to be the set of 3-planes of maximal (or minimal) energy in 6-space. It is a 5-dimensional submanifold of the Grasssmannian.

Corollary LABEL:SLAGring. The cohomology ring of the special Lagrangian manifold is the following truncated polynomial ring for which .

As an outcome of this algebraic topological study of Grassmannians we continue with the following application.

Corollary LABEL:G2R4SLfree. A closed orientable surface can be embedded into as a sLag-free submanifold if and only if the Euler characteristic of , .

As another application we can talk about SL-free embeddings of 3-manifolds into Euclidean 6-space.

Theorem LABEL:G3R6SLfree. Let be a closed, oriented 3-manifold and be an immersion, then the image of the normal bundle under the normal Gauss map is contractible and the normal bundle of the immersed submanifold is trivial, the immersion is generically special Lagrangian free.

There is a growing interest in Grassmannian manifolds due to their role in calibrated geometries. Interested reader may consult to [coassf] for an example. See also [cp5] for cohomology of Grassmannians. This paper is organized as follows. In section §2 we deal with the related Stiefel manifolds, in section §3 with some homotopy theory, in section §4 with the Grassmann manifold, in section §LABEL:secslag with the special Lagrangian submanifold, in §LABEL:secring with the cohomology ring, and finally in §LABEL:secslfreedim2-LABEL:secnormalbundle with some geometric applications.

Acknowledgements. We would like to thank M. Kreck for useful discussions, K. Mohnke for motivating us to work on this problem, in particular pointing out the paper [haefliger], and İ. Ünal. The first author would like to thank his father and family for their support during the course of this paper. This work is partially supported by the grant 114F320 of Tübitak 111Turkish science and research council..

2 Stiefel Manifolds

In order to compute the invariants of Grassmannian manifolds, some knowledge about the related Stiefel manifolds is necessary. That is why we are going to study these manifolds in this section. We start with a simpler one. Namely , the bundle of ordered orthonormal 2-frames in the Euclidean 7-space. We start with the following proposition.

Proposition 2.1.

The homology of the Stiefel manifold is the following.

Proof.

Using the cellular decomposition of Stiefel manifolds, a proof of this fact is presented at [hatcherat]. For warming up purposes for the following cases we present a different proof here. The initial homotopy groups since our 7-dimensional Stiefel manifold is 5-2-1=2-connected and [paechter1956]. Consequently, by the Hurewicz theorem we determine the homology groups upto the 3-rd level. As the next step, we are going to use the following fibration

(1)

for the rest of the homology groups. The homological Serre spectral sequence of the fibration (1) is defined together with the description of its limit as follows.

where the abelian groups are defined through

satisfies a filtration condition,

(2)

Out of this information the prior pages of our spectral sequence reads as in Table 2.

Table 2: Homological Serre spectral sequence for .

Since we know that coupling with the information we receive that . Continuing in this direction we finally reach at . The equality revealing the limit

determines the nature of the differential that is multiplication by 2, which forces . That was the only missing part of the limit page. Summing up the south-east diagonals in the limit page gives the answer. ∎

Next, using the information coming out of this proposition, we are going to manage a higher Stiefel manifold . We have the following result on this manifold.

Proposition 2.2.

The homology of the Stiefel manifold is the following.

Proof.

The Stiefel manifold is by definition equal to the set of 3-frames in 6-space. Projection onto the first vector gives the following fiber bundle (3) with fiber .

(3)

Although we can use the homology version as well, we are going to use the cohomological Serre spectral sequence related to this fiber bundle (3) to be able to use it for cup product calculations as well. We define it with the description of its limit as follows.

where abelian groups form a filtration that satisfies

(4)

This sequence behave appropriately because the base manifold is simply connected. There exists also homomorphisms called the differential maps such that

This spectral sequence converges immediately and illustrated on the Table 3.

Table 3: Cohomological Serre spectral sequence for .

The only potentially non-trivial differential is the following.

which is zero because it maps torsion to a free space. So that we have the limit as well at from the beginning. Accumulating the groups in the south-east direction again and using Poincaré duality yields the result. ∎

Next, we are going to compute the cup products, before which we need a Lemma.

Lemma 2.3.

Considering the fiber bundle (3) the following pullback maps

induced by an embedding onto some fixed fiber over a point of the base and the projection of the total space are surjective and injective, respectively.

Proof.

These pullback maps correspond to the following [hajimesato] natural maps and compositions in the spectral sequence.

Realize that the first map is a quotient map other than the identifications so that it is surjective. The second map is an inclusion in the filtration (4) other than the identifications, hence an injection. ∎

We can summarize the Propositions 2.1 and 2.2 in terms of cohomology as follows.

Corollary 2.4.

The cohomology ring of the Stiefel manifolds are the following for which .

Proof.

The first ring is obtained out of the dimensional restrictions. To deal with the second ring, after the dimensional regulations, we finally have to determine the fate of the top dimensional graded element

For this purpose we will use the Leray-Hirsch theorem [hatcherat, spanier] which gives an isomorphism on the rational cohomology. Only crucial hypothesis is what we to proved in the previous Lemma 2.3 that the cohomological pullback map from the total space to the fiber of the fibration is a surjective map. Then by the theorem we have an isomorphism between the product of fiber and the base and the total space as follows.

Restricting this isomorphism to the 12-th grading gives us the isomorphism

which means that our product has to be a generator. ∎

3 Some Homotopy Theory

In this section we are going to compute some of the homotopy theoretic invariants of the Grassmannian manifold. We need the following fiber bundle to find homotopy groups of the Grassmannian.

(5)

Exploiting this fibration, we lead to the following result.

Lemma 3.1.

The preliminary homotopy groups of the Grassmannian are the following.

Proof.

We apply the homotopy exact sequence to the fiber bundle (5), a part of which is as follows.

(6)

We use the fact that is 2-connected and Proposition 2.2 for the Stiefel manifold, and higher homotopy groups of is the same as of its universal cover which is the 3-sphere. Surjectivity of the map reveals that the only option for the 3rd level is other than the trivial group. Since the Grassmannian is 1-connected, the Hurewicz homomorphism

is an epimorphism by [hatcherat] at this level. This implies that is the only nontrivial option for the 3rd homology as well. One can continue to analyse this exact sequence by inserting the homotopy groups of the Stiefel manifold from [paechter1956] starting from the 4-th level as we did above.

We need to work with another fibration involving special orthogonal groups which is used to define the Grassmannian as well,

(7)

The homotopy sequence of this fibration at the 3rd level reads as the following,

The homotopy groups of the special orthogonal groups can be deduced from orthogonal fibrations, and injects into which makes it a subgroup of a free group hence itself free. This with (3) resolves the 4-th level. Cokernel has to be or . After this we can turn the problem into matrices. The group is a smooth 15-dimensional manifold. According to the standard embedding, in its 3x3 block lies a copy of . We need to understand the following map between the integers.

(8)

So, up to a sign, this homomorphism either maps the generator to a generator or maps to twice the generator of 3rd homotopy of . We claim that this map is multiplication by 2 upto sign, so that the embedded does not generate the 3rd homotopy group of , rather only the even members. To prove this statement let us first understand the map,

(9)

through using the fiber bundle of Stiefel manifolds,

The homotopy exact sequence at the 3rd level reveals the following,

(10)

Here, parametrizes the real 4-bundles or equivalently -bundles over as follows. Considering the atlas consisting of the two charts produced by taking out the north or the south pole. Consider the local trivializations

projecting to these contractible spaces. The transition function restricted to the above of the equivator is

which acts as for a function called the clutching function. Free homotopy type of the clutching functions classify 4-bundles upto isomorphism. is doubly covered by via the conjugation map of unit quaternions. Invariance of higher homotopy groups under covering projections imply the above equivalence. Alternatively, the homeomorphism can also be used if one does not care for the group structure. Fixing a vector and defining a map by we define the Euler class of the bundle defined by the degree of the map . So this is the way to define the map from middle to the right in the above homotopy sequence. By [walschapeuler] upto a sign, this is consistent with the definition of the Euler class as a characteristic class.

On the other hand, the vertical map in (10) is surjective, for example because of the 7-spheres of [milnors7spheres]. One can show that the Euler class for the quaternionic Hopf bundle on the 4-sphere is the generator though the Euler class for the tangent space is the twice of the generator, see [braddellthesis] for an explanation. So that . Since the kernel is free and the map is non trivial, this element generates the kernel, hence the image of .

The 3rd homotopy group of the special orthogonal groups,

reveals the fact that the half-Pontrjagin map from the stable homotopy group,

gives an isomorphism with integers. The Pontrjagin classes for spheres are trivial although the total Pontrjagin class for the quaternionic bundle is . So

which generates the image of of (8) and because of the stability generates the image of of (9). ∎

Remark 3.2.

We have found the fourth homotopy group from the above fibration (7). In [gluckmackenziemorgan] some of the rational homotopy grous are computed and the fourth level generator is the PONT manifold which is calibrated by the first Pontrjagin form.

4 The Grassmannian Manifold

In this section we are going to compute other algebraic topological invariants of the Grassmannian manifold. Akbulut and Kalafat computed [atg2] homology groups of various Grassmann bundles. We will use similar techniques to compute homology groups of At the preliminary levels we can use the homotopy theory in the previous section. Lemma 3.1 combined with the Hurewicz isomorphism we obtain,

We continue with the following central preliminary result.

Lemma 4.1.

The homology group is trivial.

Proof.

In order to prove our result, we assume that this homology groups is nontrivial. So far, after making this assumption, we are able to handle the homology upto the third level. The free homology is obtained through the Poincaré polynomial [ghv, gluckmackenziemorgan]

(11)

Various forms of the universal coefficients theorem [hajimesato] coupled with the Poincaré duality may be used to compute the higher level torsion as follows.

Similarly and . Moreover, from comparing the torsion parts of the equality for the next case,

we get . Collecting the results obtained from the assumption, the homology groups of the oriented Grassmann manifold hypothetically has to be the following sequence.

To be able to raise a contradiction, we need to define the cohomological Serre spectral sequence related to the fiber bundle (5) with the limit as follows.

(12)

where abelian groups form a filtration that satisfies

(13)

This sequence behave appropriately because the base manifold is simply connected. There exists also homomorphisms called the differential maps such that

Then keeping in mind for the fiber that,

(14)

we figure out the second page of the spectral sequence as can be seen on Table 4.

Table 4: Hypothetical cohomological Serre spectral sequence for

Because of the vanishing of the cohomology of the Stiefel manifold, the filtration at the 6-th level degenerates.

So that the vanishing of the term forces the vanishing of the limit . This is possible by a trivial kerneli so through the injectivity of the differential,

On the other hand, the domain of this differential can be computed by

which evidently can not inject into . ∎

Now, we are ready to prove the main result of this section. By further applications of Serre spectral sequence we obtain the following list.

Theorem 4.2.

The homology of the oriented Grassmann manifold is given by

Proof.

Using the homotopy and Hurewicz theorem together with Lemma 4.1, we handle the homology upto the third level. The free homology is obtained through the Poincaré polynomial (11). Various forms of the universal coefficients theorem [hajimesato] coupled with the Poincaré duality may be used to compute the higher level torsion as follows.

Similarly and . Moreover, from comparing the torsion parts of the equality for the next case,

we get . Hence the 4-th level torsion subgroup is the only remaining case. To handle this case we again define the cohomological Serre spectral sequence (12) related to the fiber bundle (5). This sequence behave appropriately because the base manifold is simply connected. After various applications of the universal coefficients theorem like,

we figure out the second page of the spectral sequence as can be seen on Table 5.

Table 5: Cohomological Serre spectral sequence for

The only missing part is the torsion at the fourth level. To address this problem we analyse the filtration at the 6-th level, so that we have the following,

The differential is injective because of the vanishing of the 6-th diagonal

which forces the group to vanish. ∎

Alternatively, one can work with the defining standard fibration (7) involving special orthogonal groups to obtain similar topological information.

Remark 4.3.

Another alternative is the following useful double fibration.

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