Functorial homotopy decompositions of iterated loop suspensions 2

FUNCTORIAL HOMOTOPY DECOMPOSITIONS OF ITERATED LOOP SUSPENSIONS YUAN ZIHONG (M.Sc., NKU, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2014 To my parents Acknowledgements I would like to express my deepest appreciation to professor Wu Jie for his inspiration, excellent guidance, support and encouragement. Without his kindest help, this thesis and many others would have been impossible. I would like to thank professor Wang Xiangjun at Nankai University who led me to algebraic topology and provided me kind encouragement and constructive suggestions for my research. Special thanks are given to Dr. Ji Feng, Dr. Ye Shengkui and Dr. Zhang Wenbin, who formed a discussion group on algebraic topology and invited me to join. The discussions enriched my knowledge and skills in research. I appreciate their cordial support on my PhD study. I also would like to express my gratitude to my friends Dr. Chen Weidong, Dr. Gao Man, Dr. Wang Yi, Wang Nan and Du Zhikun. Their kind assistance and friendship have made my life in Singapore easy and colorful. I would like to thank my friend Li Xuefang for her care, encouragement and support. v vi Acknowledgements Last but not least, I would thank my family members for their support, understanding, patience and love during past several years. Contents Acknowledgements Summary v ix Introduction 1.1 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries 2.1 2.2 Natural coalgebra retracts of tensor algebras . . . . . . . . . . . . . . 9 2.1.1 Natural k-linear transformations and graded cases . . . . . . . 11 2.1.2 General properties and special examples . . . . . . . . . . . . 15 2.1.3 Lie(n) and k(Σn )-projective submodule . . . . . . . . . . . . . 22 2.1.4 Natural coalgebra-split sub-Hopf algebras . . . . . . . . . . . . 26 Little cubes operads and homology of Ωn+1 Σn+1 X . . . . . . . . . . . 29 vii viii Contents 2.3 2.2.1 Operads and spaces associated to operads . . . . . . . . . . . 29 2.2.2 Little cubes operads and iterated loop spaces . . . . . . . . . . 33 2.2.3 Homology of Ωn+1 Σn+1 X . . . . . . . . . . . . . . . . . . . . 36 2.2.4 Homology suspensions and transgressions . . . . . . . . . . . . 43 Unstable and stable Snaith splittings . . . . . . . . . . . . . . . . . . 46 Functorial homotopy retracts of Ωn+1 Σn+1 X 55 3.1 Realizations of natural coalgebra-split sub-Hopf algebras . . . . . . . 55 3.2 Functorial homotopy retracts of Ωn+1 Σn+1 X . . . . . . . . . . . . . . 64 Further decompositions of the Snaith splitting 75 Examples and computations 83 5.1 5.2 Σt Dp X at p = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.1 Additive basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1.2 Module structures over the Steenrod algebra . . . . . . . . . . 86 5.1.3 Σt (Σ−1 Lmax ΣX) and Mp X . . . . . . . . . . . . . . . . . . . . 95 p Indecomposable submodules of general odd primes . . . . . . . . . . . 97 Bibliography 105 Summary This thesis focuses on the problem of functorial homotopy decompositions of iterated loop suspensions which is complicated and little is known. It is well known that the functorial homotopy decomposition problem of the single loop suspension is easier and has developed a lot. Motivated by the ideas developed in functorial homotopy decompositions of single loop suspensions, this thesis aims to discuss some aspects of functorial homotopy decompositions of iterated loop suspensions. The thesis consists of four parts, which are summarized below. The first part aims to introduce some basic ideas of functorial homotopy decompositions of single loop suspensions, especially the concepts related to natural algebraic decompositions of a tensor algebra. The homology of iterated loop suspensions and two versions of Snaith splittings are also incorporated. The second part analyzes the realizations of coalgebra-split sub-Hopf algebras and uses them to construct a collection of functorial homotopy retracts of iterated loop suspensions. The homology of such functorial retracts are also computed. The third part studies the Snaith splittings of the functorial homotopy retracts ix x Summary constructed in the second part. We compare these Snaith splittings with Snaith splittings of Ωn+1 Σn+1 X. For the splitting factors of Ωn+1 Σn+1 X, some functorial retracts up to certain suspensions are found. The fourth part gives examples to illustrate the Snaith splittings of the constructed functorial homotopy retracts. We compute the homology of the factor Dp X over Z/p when p = 5. The Steenrod module structures over Z/p are also calculated. For a general odd prime p, we analyze modules related to the set of indecomposable elements of H∗ Ω2 Σ2 X. 5.1 Σt Dp X at p = 5. 95 0), β. The formulas can be induced from the Adem relations: ( ) (p − 1)(b − i) − P P = (−1) P a+b−i P i ( if a < pb) a − pi i ( ) ∑ a b a+j (p − 1)(b − j) P βP = (−1) βP a+b−j P j , a − pj j ( ) ∑ a+j−1 (p − 1)(b − j) − (−1) P a+b−j βP j . + a − pj − j a b ∑ a+i (2) M2 , M3 , M4 , M5 : M2 is similar to M1 , we just check Steenrod operations on [[u, v]1 , [v, u, v]1 ]1 . For M3 , M4 , M5 , we need an additional formula: P∗r (xy) = ∑ P∗i xP∗j y. i+j=r We only need to check Steenrod actions on each generator. This is easy to compute. 5.1.3 Σt (Σ−1 Lmax p ΣX) and Mp X Let V be a k-module and let T (Lmax (V )) denote the tensor algebra generated by p (V ). It is a natural coalgebra-split sub-Hopf algebra of tensor algebra T (V ). Lmax p ΣX, which is a functorial homotopy retract Hence we can construct a space Σ−1 Lmax p ¯ ) be the sub Lie algebra of the free Lie algebra L(V ) generated by of Ω2 Σ2 X. Let L(V Ln (V ) (1 < n < p). Let V¯ be p-dimensional k-module generated by the {x1 , · · · , xp }. (V ), we have the following proposition. Recall the structure of Lmax p 96 Chapter 5. Examples and computations Proposition 5.3 ([19], Proposition 11.6). There is an isomorphism of Σp -modules: ¯ V¯ ) ∩ Lie(p). Lmax (p) ∼ = L( This implies that Lmax (V ) = [L2 (V ), Lp−2 (V )] + [L3 (V ), Lp−3 (V )] + · · · + [Lp−2 (V ), L2 (V )]. p ¯ ∗ (X; Z/p). Consider the case p = 5, Lmax (V ) has a basis Let V = H [[u, v], [u, u, v]], [[u, v], [v, u, v]]. From the discussions in Chapter 3, we know the above basis is mapped by the following map ¯ ∗ Σ−1 Lmax ΣX → H ¯ ∗ Ω2 Σ2 X i∗ : H p to [[u, v]1 , [u, u, v]1 ]1 , [[u, v]1 , [v, u, v]1 ]1 . ¯ ∗ Σt (Σ−1 Lmax ΣX), we use the same symbol to denote the corresponded basis For H under the canonical isomorphism. All these imply the following proposition. Proposition 5.4. ¯ ∗ Σt (Σ−1 Lmax ΣX) ∼ H = M2 . Since Σt Dp X ≃ Σt (Σ−1 Lmax ΣX) ∨ Mp X, p ¯ ∗ Mp X is also clear. the structure of H 5.2 Indecomposable submodules of general odd primes Proposition 5.5. There is a decomposition of right A-modules, ¯ ∗ Mp X ∼ H = M1 ⊕ M3 ⊕ M4 ⊕ M5 . ¯ ∗ Mp X is splittable, it is natural to ask that Remark 5.6. As a right A-module, H whether Mp X is splittable as a topological space. Especially, whether the functorial homotopy decomposition exists or not. 5.2 Indecomposable submodules of general odd primes For a general odd prime p, we have similar discussions. Let QH∗ Ω2 Σ2 X be the set of the indecomposable elements of H∗ Ω2 Σ2 X. Then it has an additive basis { } T1 X = ζ1k ξ1s y | y is a basic λ1 -product, k = 0, and s ⩾ . ¯ ∗ Dp X = QH∗ Ω2 Σ2 X ∩ H ¯ ∗ Dp X. Consider the canonical isomorphism Let QH ¯ ∗ Dp X → H ¯ ∗ Σt Dp X, H ¯ ∗ Dp X under the above isomorphism by QH ¯ ∗ Σt Dp X. An denote the image of QH ¯ ∗ Σt Dp X can be written as additive basis of QH {y | y is a basic λ1 -product of length p} ∪ {ξ1 u, ξ1 v, ζ1 u, ζ1 v}. 97 98 Chapter 5. Examples and computations Let A denote the Steenrod algebra. By Proposition 5.1, it is not difficult to verify ¯ ∗ Σt Dp X is closed under A-actions. Now, define that QH ¯ ∗ Σt (Σ−1 Lmax ΣX) = H ¯ ∗ Σt (Σ−1 Lmax ΣX) ∩ QH ¯ ∗ Σt Dp X, QH p p ¯ ∗ Mp X = H ¯ ∗ Mp X ∩ QH ¯ ∗ Σt Dp X. QH Then, ¯ ∗ Σt Dp X ∼ ¯ ∗ Σt (Σ−1 Lmax ΣX) ⊕ QH ¯ ∗ Mp X. QH = QH p Now let us discuss these two A-submodules. ¯ ∗ Σt (Σ−1 Lmax ΣX) is a A-submodule of QH ¯ ∗ Σt Dp X,i.e., Proposition 5.7. H p ¯ ∗ Σt (Σ−1 Lmax ΣX) = H ¯ ∗ Σt (Σ−1 Lmax ΣX). QH p p Proof. From the structure of Lmax , for a k-module V , we have p Lmax (V ) = [L2 (V ), Lp−2 (V )] + [L3 (V ), Lp−3 (V )] + · · · + [Lp−2 (V ), L2 (V )]. p An additive basis for Lmax (V ) contains all elements in the Hall set (see [16], §4.)of p form [a, b]0 , l(a) + l(b) = p, l(a), l(b) ⩾ 2. ¯ ∗ Σt (Σ−1 Lmax ΣX) (see the discussion on Hence, we have a corresponding basis for H p page 71). We just change all the λ0 -products above to λ1 -products. The basis is {[a, b]1 , l(a) + l(b) = p, l(a), l(b) ⩾ 2}. 5.2 Indecomposable submodules of general odd primes 99 This basis is a subset of {y | y is a basic λ1 -product of length p} . ¯ ∗ Σt (Σ−1 Lmax ΣX) is the homology of Finally, the Steenrod actions are closed since H p a topological space Σt (Σ−1 Lpmax ΣX). This completes the proof. ¯ ∗ Mp X is of dimension p + 3. Proposition 5.8. QH ¯ ∗ Σt Dp X. Delete the basis of H ¯ ∗ Σt (Σ−1 Lmax ΣX), Proof. Use the additive basis of H p we still have [v, · · · , v, u, v]1 , [v, · · · , v, u, u, v]1 , · · · , [v, u, · · · , u, v]1 , [u, · · · , u, v]1 , (5.3) ξ1 v, ζ1 v, ξ1 u, ζ1 v. For the elements in the first row, the commutator length are p. From left to right, the ith commutator contains i copies of u. Thus, there are p − commutators, and hence p + elements in total. ¯ ∗ Mp X is in fact the submodule M1 . Now let us compute the If p = 5, QH ¯ ∗ Mp X, and try to find a basis of it. Steenrod operations over QH ¯ ∗ Mp X. Hence, Remark 5.9. Consider the set (5.3), it may not be a subset of QH ¯ ∗ Mp X. it may not be a basis of QH ¯ ∗ Mp X: Proposition 5.10. Let a = −[v, · · · , v, u, v]1 . Then there is a basis for QH a, (P∗1 )(i) a(1 ⩽ i ⩽ p − 2), ξ1 u, ξ1 v, ζ1 u, ζ1 v. (5.4) 100 Chapter 5. Examples and computations where (P∗1 )(i) means iterated composition of i copies of P∗1 . All the nontrivial operations β, P∗1 , P∗p are shown in the following diagram: ξ1 v ❚❚❚❚ P∗1 ❚❚❚❚ ❚❚β❚❚ ❚❚❚❚ ❚❚❚❚ ) ED ζ1 v  a = −[v, · · · , v, u, v]1  P∗1 P∗1 a P∗p P∗1  .  P∗p P∗1 (P∗1 )(p−2) a = −[u, · · · , u, u, v]1 BC ξ1 u o❚❚❚❚ ❚❚❚❚ ❚❚❚β❚ ❚❚❚❚ ❚❚❚❚ )  ζ1 u. Other nontrivial actions are P∗r (2 ⩽ r ⩽ p − 1), P∗p β = βP∗p . ¯ ∗ Σt Dp X: Proof. Let us check the basis first. Consider the additive basis of QH {y | y is a basic λ1 -product of length p} ∪ {ξ1 u, ξ1 v, ζ1 u, ζ1 v}. It is easy to see that ξ1 v is of highest degree, and all the other elements have degrees less than it. Let m = |ξ1 v|. Then ¯ ∗ Σt Dp X)m ∼ (QH = Z/p. 5.2 Indecomposable submodules of general odd primes 101 The generator is ξ1 v. On the other hand, ¯ ∗ Σt Dp X ∼ ¯ ∗ Σt (Σ−1 Lmax ¯ ∗ Mp X, QH ΣX) ⊕ QH = QH p ¯ ∗ Σt (Σ−1 Lmax ΣX) (from Proposition 5.7). Hence, ξ1 v is contained in and ξ1 v ∈ / QH p ¯ ∗ Mp X. QH ¯ ∗ Mp X, apply the Steenrod operations, from Proposition 5.1, all If ξ1 v is in QH ¯ ∗ Mp X: the following p + elements are contained in QH a, (P∗1 )(i) a, (1 ⩽ i ⩽ p − 2), ξ1 u, ξ1 v, ζ1 u, ζ1 v. ¯ ∗ Mp X is p + 3, we only need to verify the above set Since the dimension of QH is linearly independent. In fact, we claim that the set {a, (P∗1 )(i) a, (1 ⩽ i ⩽ p − 2)} ∪ {ξ1 u, ξ1 v, ζ1 u, ζ1 v} (5.5) ¯ ∗ Σt (Σ−1 Lmax ΣX)} ∪ {a basis of QH p ¯ ∗ Σt Dp X. Compare this with the basis is a basis of QH {y | y is a basic λ1 -product of length p} ∪ {ξ1 u, ξ1 v, ζ1 u, ζ1 v}. The different parts are {(P∗1 )(i) a, (0 ⩽ i ⩽ p − 2)}, {[v, · · · , v, u, v]1 , [v, · · · , v, u, u, v]1 , · · · , [v, u, · · · , u, v]1 , [u, · · · , u, v]1 }. Consider the above sets. First, (P∗1 )(i) a (1 ⩽ i ⩽ p − 2) are nontrivial and not 102 Chapter 5. Examples and computations ¯ ∗ Σt (Σ−1 Lmax ΣX). This is because contained in QH p (P∗1 )(P∗1 )(i) = (P∗1 )(i+1) , (P∗1 )(p−2) a = −[u, · · · , u, u, v]1 ̸= 0, ¯ ∗ Σt (Σ−1 Lmax ΣX). − [u, · · · , u, u, v]1 ∈ / QH p Next, consider the degree of (P∗1 )(i) a, (0 ⩽ i ⩽ p − 2), we have |a| = | − [v, · · · , v, u, v]1 |, |P∗1 a| = |[v, · · · , v, u, u, v]1 |, . |(P∗1 )(p−2) a| = |[u, · · · , u, v]1 |. Denote the degrees above by a sequence of positive integers. m0 > m1 > · · · > mp−2 . Consider the homomorphism ¯ ∗ Σt Dp X → QH ¯ ∗ Σt Dp X QH ∼ ¯ t ¯ ∗ Σt (Σ(−1) Lmax ΣX) = QH∗ Σ Dp X, QH p since the quotient of each element in the set {[v, · · · , v, u, v]1 , [v, · · · , v, u, u, v]1 , · · · , [v, u, · · · , u, v]1 , [u, · · · , u, v]1 } is nontrivial, it is easy to find that ¯ ∗ Σt Dp X)m ∼ (QH Z/p. i = 5.2 Indecomposable submodules of general odd primes 103 ¯ ∗ Σt Dp X)m Consequently, we can choose arbitrary p−1 non-zero elements from (QH i (0 ⩽ i ⩽ p − 2), one for each mi , to replace [v, · · · , v, u, v]1 , [v, · · · , v, u, u, v]1 , · · · , [v, u, · · · , u, v]1 , [u, · · · , u, v]1 . ¯ ∗ Σt Dp X from the original basis: Such a process can form a new basis of QH {y | y is a basic λ1 -product of length p} ∪ {ξ1 u, ξ1 v, ζ1 u, ζ1 v}. On the other hand, under the homomorphism ¯ ∗ Σt Dp X → QH ¯ ∗ Σt Dp X QH ∼ ¯ t ¯ ∗ Σt (Σ(−1) Lmax ΣX) = QH∗ Σ Dp X, QH p the image of each element in {(P∗1 )(i) a, (0 ⩽ i ⩽ p − 2)} ¯ ∗ Σt (Σ(−1) Lmax ΣX). is also nontrivial since (P∗1 )(i) a, (0 ⩽ i ⩽ p−2) is not contained in QH p It is easy to see that we can use the set {(P∗1 )(i) a, (0 ⩽ i ⩽ p − 2)} to replace {[v, · · · , v, u, v]1 , [v, · · · , v, u, u, v]1 , · · · , [v, u, · · · , u, v]1 , [u, · · · , u, v]1 }. 104 Chapter 5. Examples and computations ¯ ∗ Σt Dp X, which is exactly This forms a new basis of QH {a, (P∗1 )(i) a, (1 ⩽ i ⩽ p − 2)} ∪ {ξ1 u, ξ1 v, ζ1 u, ζ1 v} ¯ ∗ Σt (Σ−1 Lmax ΣX)}. ∪ {a basis of QH p This finishes the proof of the basis part. Second, the diagram and all the nontrivial Steenrod actions can easily be induced from Proposition 5.1. Bibliography [1] F. R. Cohen. Combinatorial group theory in homotopy theory, I. preprint, http://www.math.rochester.edu/people/faculty/cohf/. [2] F. R. Cohen. 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FUNCTORIAL HOMOTOPY DECOMPOSITIONS OF ITERATED LOOP SUSPENSIONS YUAN ZIHONG NATIONAL UNIVERSITY OF SINGAPORE 2014 Functorial homotopy decompositions of iterated loop suspensions Yuan Zihong 2014 [...]... 2 2 in [29 ] (2) Homology of functorial homotopy retracts If we have a functorial decomposition of ΩΣ ΩΣ X ≃ A(X) × B(X), then A(X) is a functorial homotopy retract of ΩΣ X By applying iterated loop functor, Ωn+1 Σn+1 X ≃ Ωn A(Σn X) × Ωn B(Σn X) We can see that Ωn A(Σn X) is a functorial homotopy retract of Ωn+1 Σn+1 X For each natural coalgebra retract of T (V ), we have the related functorial homotopy. .. (V ) can induce a functorial homotopy decomposition of ΩΣ X In the later development in [17, 18, 25 , 20 ], the problem was generalized to functorial homotopy decomposition of loop functor Ω over co-Hspaces In [18], it showed that for a p-local simply connected co-H-space of finite type, any natural coalgebra decomposition of T (V ) implies a functorial homotopy decomposition of the loop functor Ω The... retract of Ωn+1 Σn+1 X in this way Compute the homology of them and characterize functorial homotopy retracts of this type (constructed from functorial retracts of ΩΣ) (3) Ωn Amin (Σn X) Can Ωn Amin (Σn X) be decomposed functorially further? In other words, whether this functorial homotopy retract is a minimal homotopy retract of Ωn+1 Σn+1 X or not Note that Amin (X) is a minimal functorial retract of ΩΣ... CW-complexes unless otherwise stated Chapter 2 Preliminaries 2. 1 Natural coalgebra retracts of tensor algebras We first introduce the concept of natural coalgebra retracts This is one of the key concept in the functorial homotopy decompositions of the loop suspension functor Let M (V ) be a functorial k-submodule of T (V ) In other words, M is a subfunctor of T from k-modules to k-modules Denote the... constructed functorial retracts They are compared to the Snaith splittings of iterated loop suspensions When the natural coalgebra-split sub-Hopf algebra is T (Lmax (V )), we find some functorial m retracts Theorem 1 .2 There is an integer t such that Σt Dj (Σ-n Lmax Σn X) is a functorial m homotopy retract of Σt Djm (X) 1.1 Organization of this thesis This thesis is organized as follows Chapter 2 First,... homology of Ωn+1 Σn+1 X Finally, we give the Snaith splitting of Ωn+1 Σn+1 X, and also some the unstable results related to this splitting Chapter 3 We construct functorial homotopy retracts of Ωn+1 Σn+1 X from natural coalgebra-split sub-Hopf algebras of tensor algebra, and discuss the homology of them Chapter 4 We consider the Snaith splittings of the functorial homotopy retracts constructed in Chapter 2. .. [19], there exist a subfunctor Lmax of Ln and a n submodule Liemax (n) of Lie(n) with properties: Proposition 2. 14 ([19],§6-§7) (1) Liemax (n) is a k(Σn )-projective submodule of Lie(n), any k(Σn )-projective submodule of Lie(n) is a k(Σn )-retract of Liemax (n) (2) Lmax is a Tn -projective subfunctor of Ln , any Tn -projective subfunctor of Ln is n a functorial retract of Lmax n (3) There exists an element... → Amin (V ; M n ) is the identity 0 (2) p(x1 ⊗ · · · ⊗ xm ) = (((x1 · x2 ) · x3 ) · · · · xm ) for m ⩾ 1 and xj ∈ V , where · is the multiplication in Amin (V ; M n ) 22 Chapter 2 Preliminaries p is a functorial morphism of coalgebras and the composition s min p A Amin (V ; M n ) − − T (V ) → Amin (V ; M n ) −→ − is a functorial isomorphism of coalgebras [19, p 32] Define B max (V ; M n ) = k □Amin... Proposition 2. 9, we have B max (V ; M n ) is also a natural coalgebra retract of T (V ) The following is the main properties of B max (V ; M n ) Proposition 2. 11 ([19], Proposition 6.1) (1) B max (V ; M n ) is a natural coalge- bra retract of T (V ) (2) B max (V ; M n ) is a natural sub-Hopf algebra of T (V ) (3) There is a functorial isomorphism of coalgebras k ⊗B max (V ;M n ) T (V ) ∼ Amin (V ; M n ) = 2. 1.3... a functorial retract of ΩΣ X and the functorial injection jX : ΩΣ Q(X) → ΩΣ X is a loop map (2) Ωn+1 Σn+1 (Σ−n QΣn X) is a functorial homotopy retract of Ωn+1 Σn+1 X The functorial inclusion Ωn jΣn X : Ωn+1 Σn+1 (Σ−n QΣn X) → Ωn+1 Σn+1 X is a Ωn+1 -map ¯ (3) Let V = H∗ (X), then H∗ (Q(X)) ∼ Q(V ), H∗ (ΩΣ Q(X)) ∼ B(V ), and = = H∗ (Σ−n QΣn X) ∼ s−n Q(sn V ) = We also consider the Snaith splittings of . . . 29 vii viii Contents 2. 2.1 Operads and spaces associated to operads . . . . . . . . . . . 29 2. 2 .2 Little cubes operads and iterated loop spaces . . . . . . . . . . 33 2. 2.3 Homology of Ω n. [29 ]. (2) Homology of functorial homotopy retracts. If we have a functorial de- composition of ΩΣ ΩΣ X ≃ A(X) × B(X), then A(X) is a functorial homotopy retract of ΩΣ X. By applying iterated loop. aspects of functorial homotopy decompositions of iterated loop suspensions. The thesis consists of four parts, which are summarized below. The first part aims to introduce some basic ideas of functorial