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Algebraic Structures Based on a Classifying Space of a Compact Lie Group
We analyze the algebraic structures based on a classifying space of a compact Lie group. We construct the connected graded free Lie algebra structure by considering the rationally nontrivial indecomposable and decomposable generators of homotopy groups and the cohomology cup products, and we show that the homomorphic image of homology generators can be expressed in terms of the Lie brackets in rational homology. By using the Milnor-Moore theorem, we also investigate the concrete primitive elements in the Pontrjagin algebra.
A Lie group is a differentiable manifold with a group structure in which the multiplication and the inversive map () are differentiable. Therefore, it can be studied using differential calculus in contrast with the case of more general topological groups as a special case of H-spaces. It is well known that the only spheres that are connected H-spaces are , , and . We note that the first two spheres are Lie groups while the last one is just an H-space which is not an -space but just an -space in the sense of Stasheff . Lie groups play an enormous role in algebraic topology as well as modern differential geometry on several different levels. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Moreover, linear actions of Lie groups are especially important and are studied in representation theory.
As usual we let and be the suspension and loop functors in the (pointed) homotopy category, respectively. It is well known that the functors and are examples of adjoint functors. Moreover, co-H-spaces and H-spaces are important objects of research in homotopy theory and they are the dual notions in the sense of Eckmann and Hilton. We refer to Arkowitz’s paper  and Scheerer’s article  for a survey of the vast literature about co-H-spaces, H-spaces, and related topics.
Let denote the set of all homotopy types such that and have the same -type for each nonnegative integer (see [4–6]). McGibbon and Møller  showed that if is a connected compact Lie group, then its classifying space usually has an uncountable except for several cases and gave an excellent set of examples. Furthermore, in  the classical projective -spaces (real, complex, and quaternionic) were studied in terms of their self-maps from a homotopy point of view. Recently, some common fixed point results for single as well as set valued mappings involving certain rational expressions in complete partial metric spaces were obtained in . Moreover, some fixed point and common fixed point theorems on ordered cone -metric spaces were also established in .
In this paper all spaces are based and have the based homotopy type of based, connected CW-complexes. All maps and homotopies preserve the base point. Unless otherwise stated, we do not distinguish notationally between a map and its homotopy class.
The main purpose of this paper is to investigate the algebraic explanation based on a classifying space of the compact Lie group . After constructing self-maps using the suspension structure, we define a useful commutator of self-maps on the suspension of a classifying space of the compact Lie group. We construct the connected graded free Lie algebra structure by considering the rationally nontrivial indecomposable and decomposable generators of homotopy groups and the cohomology cup products. We show that the homomorphic image of homology generators can be expressed in terms of the Lie brackets in rational homology. By using the Milnor-Moore theorem, we also investigate the concrete primitive elements as the images of the Hurewicz homomorphisms in the Pontrjagin algebra.
Let , , , and denote the ring of integers, the fields of rational, real, and complex numbers, respectively. The unitary group is the group of unitary matrices with the group operation that of matrix multiplication as a subgroup of the general linear group . The unitary group is a real Lie group of dimension . The Lie algebra of consists of skew-Hermitian matrices with the Lie bracket given by the commutator. The first unitary group is canonically the circle group consisting of all complex numbers with absolute value 1 under multiplication; that is, the set of all unitary matrices. The first unitary group shows up in a variety of forms in mathematics. Some of the more common forms are as follows: where is a one torus and is the orthogonal matrices with determinant . The latter is said to be the special orthogonal group.
A classifying space of a topological group is the quotient of a weakly contractible space by a free action of . It has the property that any principal -bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle . We note that the classifying space functor is essentially inverse to the loop space functor in algebraic topology.
If is a co-H-group with comultiplication and homotopy inverse, then for every pointed space the set of homotopy classes from to can be given the structure of a group under the addition as follows: for , the binary operation, denoted “”, is defined by the homotopy class of the composition of maps where is a folding map.
The Eckmann-Hilton dual of a co-H-group is an H-group (see [11, 12]). As an adjointness, if is any pointed space and is an H-group, then the set becomes a group if we define the product “” to be the homotopy class of the composition of maps where is the diagonal map, , and is a multiplication.
The principle examples of a co-H-group and an H-group are the suspension and the loop space of a space , respectively.
We note that has a CW-decomposition as follows: where is an attaching map for .
From now on, we denote by the -skeleton of a CW-complex . We define maps for as follows.
Definition 1. The cofibration sequence induces an exact sequence of groups for each . We now take an essential map for each . Similarly, by using the above exact sequences, we can choose an essential map such that for .
In the above definition, we note that , and thus is an isomorphism. We also note that
We now define the following.
Definition 2. We define a rationally nontrivial homotopy element of the homotopy groups modulo torsions by for each .
Now we take the self-map by the adjointness of for each .
We recall that as a graded -module, where is the standard generator of for each . The salient Bott-Samelson theorem  says that the Pontrjagin algebra is isomorphic to the tensor algebra . In other words, the rational homology of is the tensor algebra generated by , where is a rational homology generator with diagonal and . Here, is the canonical inclusion map defined by , and means . From the Serre spectral sequence of a fibration , we have an algebra isomorphism Here is the polynomial algebra over generated by of degree ; that is, is a generator of with the Kronecker index .
We now consider a wedge of spheres for . Let be the th inclusion for . We then inductively define and order basic (Whitehead) products as follows. Basic products of weight 1 are (in order) . Assume basic products of weight < have been defined and ordered so that if , any basic product of weight is less than all basic products of weight . Then a basic product of weight is a Whitehead product , where is a basic product of weight and is a basic product of weight , , . Furthermore, if is a Whitehead product of basic products and , then we require that . The basic products of weight are ordered arbitrarily among themselves and are greater than any basic product of weight < . Note that to a basic product of weight we can associate a string of distinct symbols , for , which are the elements which appear in the basic products. Suppose in the basic product , occurs times, . Then the height of the basic product is and the length is . Clearly if has height , then .
We end this section with the following Hilton’s formula .
Theorem 3. Let the ordered basic products of be with the height of . Then for every , The isomorphism is defined by
We note that the direct sum is finite for each since .
3. Commutators and Lie Algebra Structures
By using the addition of a co-H-group in Section 2, we define the following.
Definition 4. We define a commutator
of and in by
where the operations are the suspension additions on , and
is the suspension inverse defined by
for and .
Let be the map of loop inverse given by , where , .
Similarly, we define the following.
Definition 5. The map is a commutator of and in defined by where , and the multiplication is the loop multiplication.
Remark 6. Let be a map given by Then we get where is the diagonal map.
Note that the weak category of is not finite because there are infinitely many nonzero cohomology cup products in it, and thus it has the infinite Lusternik-Schnirelmann category [12, Chapter X]. Moreover, Arkowitz and Curjel [15, Theorem 5] showed that the -fold commutator is of finite order if and only if all -fold cup products of any positive dimensional rational cohomology classes of a space vanish (see also ). Therefore, we can consider the iterated commutators which are nontrivial in .
Let and be the first and second inclusions between based spaces, respectively; that is, and , where is the base point of . Recall that an element is said to be primitive if and only if in homology, where is the diagonal map.
Theorem 7. If is a simply connected topological space and if is a field of characteristic zero, then (1) the Samelson product makes into a graded Lie algebra denoted by ; (2) the Hurewicz homomorphism for is an isomorphism of onto the Lie algebra of primitive elements in ; (3) the Hurewicz homomorphism extends to an isomorphism of graded Hopf algebras , where is the universal enveloping algebra of .
It is natural to ask what are the rationally nontrivial indecomposable and decomposable generators of the graded Lie algebra for ? The following gives an answer to this question.
Theorem 8. The connected graded Lie algebra for with the Samelson products modulo torsions is as follows: where the dimension of is equal to in the graded homotopy groups for each .
Proof. It suffices to show that the iterated Samelson products in homotopy groups are rationally nontrivial decomposable generators, where the are indecomposable generators in dimension , for and .
We first note that the Eckmann-Hilton dual of the Hopf-Thom theorem (see [11, pages 263–269] and [12, Chapter III]) says that has the rational homotopy type of the wedge products of infinitely many spheres; that is,
Let be the adjoint of . We prove the result in the case of twofold Samelson products. Suppose that is rationally trivial in . It follows by the adjointness and the Hilton’s formula that the twofold Whitehead products have a finite order in which is a subgroup of . By using a cofibration sequence we have where is the attaching map. The assumption shows that when is rationalized, and thus where “” is a rational homotopy equivalence. We also have a contradiction by applying the cohomology cup products to the above rational homotopy equivalence.
For induction, we now suppose that the -fold Samelson products are rationally nontrivial in the homotopy group . By Theorem 3 and adjointness again, we can consider the iterated Whitehead product as a rational generator of
Similarly, a cofibration shows that where is the attaching map. If this map has a finite order in homotopy groups, then we have a contradiction again by the same argument of the cohomology cup products.
Finally by taking the adjointness, we complete the proof.
Let be the composition of the rationally nontrivial indecomposable element of with the rationalization for each . Since there is a one-to-one correspondence between the integral generators of homotopy groups modulo torsions and rational generators of rational homotopy groups; that is, , by using the Milnor-Moore theorem and Theorem 8, we have the following.
Corollary 9. The graded rational homotopy group with the Samelson products becomes a connected graded free Lie algebra generated by ; that is,
We note that the iterated Samelson products are decomposable generators in the above free Lie algebra.
It is well known that the Hurewicz homomorphism carries the Samelson product into the Lie bracket defined by where , and the multiplication is the Pontrjagin multiplication.
Theorem 10. Let be the adjoint of . Then one has the following: (1); (2), where is a rational homology generator in dimension , is the Lie bracket, and and are homotopy elements of .
Proof. (1) The adjointness shows that, for , , , and , the map
is a group isomorphism. It thus follows that
which is the commutator of and in .
(2) For the second part, if is the projection map to the top cell and if is the Samelson product in , then the following diagram is commutative up to homotopy (see also [20, Theorem 1.4]): (40)
By applying the rational homology to the above diagram, we have
Here, (i) is also used as a generator of (ii); (iii) is an isomorphism sending the generator to the fundamental homology class .
Theorem 11. Let be the canonical inclusion and let be the projection map. Then for a given commutator , there exists a map such that .
Proof. We first show that the following diagram is strictly commutative:
Indeed, the composition induces a map sending to
On the other hand, sends to
By adjointness, we have where is an element of . Therefore, we get
We now consider the cell structure on with
Since the restrictions and are inessential from our construction of for each , we see that if or , then is null homotopic. By using the homotopy extension property, we can extend the null homotopy to all of . Thus, by the cellular approximation theorem, we have where is a cellular map. Therefore there exists a map such that the following diagram is commutative up to homotopy: (54) where the top row is a cofibration sequence as required.
The Milnor-Moore theorem asserts that the image of the Hurewicz homomorphism is primitive. The following is another expression of the primitive elements in the Pontrjagin algebra.
Theorem 12. The image of a homomorphism is primitive for each .
Proof. Since , can be factored as so that the restriction to the bottom sphere of the map coincides with the map , and since is -connected, by the Hurewicz isomorphism theorem, every class in is spherical, and thus primitive. We therefore know that the image of lies in the set of primitives in , so does the image of for each .
Since the restriction to the skeleton is null homotopic, by Theorem 12, we have the following.
Corollary 13. is in the Lie subalgebra of primitive elements in .
Remark 14. We note that the self-map is a loop map; thus it is an H-map. It is well known in [3, page 75] that there is a bijection between the groups and of homotopy classes of H-maps . We thus get the corresponding self-map in the group . Moreover, by using the classical Whitehead theorem, we obtain a homotopy self-equivalence of the form ; that is, , where is the identity map and “+” is the suspension addition on .
The author is grateful to an anonymous referee for a careful reading and many helpful suggestions that improved the quality of the paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0007611).
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Copyright © 2013 Dae-Woong Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.