Permutohedra, Configuration Spaces and Spineless Cacti Yongheng Zhang Purdue University

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This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. Graduate School Form 30 Updated 1/15/2015

PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance

This is to certify that the thesis/dissertation prepared

By Yongheng Zhang

Entitled Permutohedra, Configuration Spaces and Spineless Cacti

For the degree of Doctor of Philosophy

Is approved by the final examining committee:

Ralph Kaufmann Chair James McClure

David B. McReynolds

David Gepner

To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of Integrity in Research” and the use of copyright material.

Approved by Major Professor(s): Ralph Kaufmann

Approved by: David Goldberg 4/22/2015 Head of the Departmental Graduate Program Date

PERMUTOHEDRA, CONFIGURATION SPACES AND SPINELESS CACTI

A Dissertation

Submitted to the Faculty

Purdue University

Yongheng Zhang

In Partial Fulﬁllment of the

Requirements for the Degree

Doctor of Philosophy

May 2015

Purdue University

West Lafayette, Indiana ii

ACKNOWLEDGMENTS

The earliest inspiration for the results mentioned in this paper came from a picture (a cylinder with two triangles attached) Ralph Kaufmann showed me as an interesting example of a simplical complex before he became my advisor more than four years ago. Throughout these years, he has generously shared his knowledge, enthusiasm, wisdom and vision with me, and this work would have remained at most a conjecture without his continual guidance, suggestion and support. I’m most grateful to him not only for his role as an advisor but also because he has been a constant source of broader intellectual inspiration for me. I am also grateful to James McClure from whose work I learned a myriad of topology tools and the context of my work. Having been both grader and course assistant for him proved to be precious experience in which I learned how much time and energy one has to devote in order to create an excellent course. My sincere thanks also go to Ben McReynolds for his course on Rie- mannian geometry and his pivotal career advice to me. Even though David Gepner is the last committee member I met, I learned so much from him, not just homotopy theory, but also how to be a scholar.

IwouldalsoliketothankLaszloLempertforrecommendingbooksonspectral sequences while I was taking his course on several complex variables so that I began to understand the homology of the little 2-discs operad, and William Heinzer for telling me THAT IS THE FUN PART!! when he heard that I was stuck on my research.

I’m also grateful to my colleagues, namely Ralph Kaufmann’s other students: Ben Ward, Byeongho Lee, Yu Tsumura, Jake Noparstak, Jason Lucas and Artur Jackson, for enlightening conversations. I ﬁrst heard the word operad from Ben in a graduate topology conference at Michigan State University, and then Jason and I started to iii work on it. I am most indebted to Jason for countless hours of discussions and sharing his drawings of circles and squares which I also produced many.

I would also like to thank all the professors and mentors, both in and out of the math department, from whom I have taken so many excellent courses and received valuable advice on teaching. In particular, I would like to thank Steven Bell for pulling out my graduate school application possibly from a pile of discarded ﬁles knowing I drove a broken car on a blizzard February morning from Hammond to West Lafayette.

Ialsowanttothankmyroommates,ocemates, teammates on the soccer ﬁeld, and all my old and new friends for making my life colorful. Finally, I dedicate this thesis to little T. and his mom. iv

TABLE OF CONTENTS

Page LIST OF TABLES ...... vi LIST OF FIGURES ...... vii SYMBOLS ...... x ABSTRACT ...... xii 1 INTRODUCTION ...... 1 1.1 Motivation ...... 1 1.2 Results ...... 4 1.3 Outline ...... 5 2 THE LITTLE 2-DISCS AND THE SPINELESS CACTI OPERADS ... 6 2.1 Operads and their morphisms ...... 6 2.1.1 The little 2-discs operad D2 ...... 9 2.1.2 The spineless cacti operad Cact ...... 11 2.2 Dimensions ...... 15 3 PERMUTOHEDRA ...... 21 3.1 Permutohedra as convex hulls ...... 21 3.2 Permutohedra as realizations of posets ...... 23 4 PERMUTOHEDRAL STRUCTURE OF F (R2,n) ...... 25 4.1 The poset n ...... 25 4.2 A theoremJ of Blagojevi´cand Ziegler ...... 26 5 PERMUTOHEDRAL STRUCTURE OF Cact1(n) ...... 27 5.1 Reformulation of Cact1(n)astherealizationofaposet ...... 27 1 5.2 Permutohedra Pn in Cact (n)...... 29 5.3 Permutohedral structure in Cact1(n)inothercontexts ...... 36 5.4 The spaces (n) ...... 40 C 6 HOMOTOPY EQUIVALENCES BETWEEN (n)and (n) ...... 44 6.1 Extended products of permutohedra andF a simplex C ...... 45 6.2 Iterated cone construction ...... 49 6.3 The homotopy ...... 51 REFERENCES ...... 56 A Proof of Proposition 6.1.1 ...... 59 v

Page BProofofProposition6.1.2 ...... 66 C How were the results discovered? ...... 68 C.1 A Motivating Example ...... 68 1 C.2 Generators of H3( (3)) and H3(Cact (3)) ...... 70 C.2.1 (n)andCactF 1(n), n =1, 2, 3 ...... 70 C.2.2 ComputationF and comparison of the second homology generators...... 74 C.3 Homotopy equivalences f : Cact1(3) (3) : g...... 76 C.4 Do not start to do it unless you think$ itF is obvious! ...... 78 VITA ...... 84 vi

LIST OF TABLES

Table Page

2.1 Ranks of Chen(PBn)k and Chen(F1 Fn 1)k, where k occupies the top row...... ··· 17 vii

LIST OF FIGURES

Figure Page

2.1 A little 2-discs conﬁguration determined by an element in D2(5). .... 10

2.2 Action of S5 on D2(5)...... 10

2.3 An example of operadic insertion 2 : D2(3) D2(2) D2(4). The dashed circle is not part of the picture. ...... ⇥ ! 11 2.4 An element in Cact(5): it is the isotopy class of orientation and intersection parameter preserving embeddings of the five standard circles of radii 37 3 2, 3, 30 , 1, 2 such that the images form a rooted planted tree-like configuration of circles. The local zeros are denoted by black dots. The global zero is denoted by a black square...... 14 2.5 An example of operadic insertion : Cact(3) Cact(2) Cact(4). .. 14 2 ⇥ ! 3.1 The permutohedra P1, P2 and P3. Each vertex is labelled on the left and v on the right...... 22 2 3.2 The codimension 1 faces of P4 and their indexing elements in 1234. Visible faces are labelled by bold-faced numbers. The faces of theJ types abc d, | ab cd and a bcd are anely isomorphic to P3 P1, P2 P2 and P1 P3, respectively.| | ...... ⇥ ⇥ ⇥ 24 5.1 Example of four elements of and their morphisms ...... 27 Tn ! 5.2 A subposet of and its image under the functor C...... 29 Tn 5.3 Examples of trees in and the associated cacti pictures...... 30 T53214 5.4 The sets 2 , 1 , 0 ...... 31 T572 T572 T572 5.5 An example of grafting of trees and the associated cacti picture. .... 32 5.6 The elements of 5 [l] where l = l , l and l =36,l =214...... 33 T532146 1 2 1 2 5.7 colim C, i =1, 2, 3...... 35 T1234,i 5.8 Let c be an element in Cact1(5) shown on the left. Let ⇤(3, 5) be 2 defined by (1) = 5, (2) = 1 and (3) = 2. Then to get ⇤(c), we first contract lobes labelled by [5] 5, 1, 2 and then relabel 5 as 1, 1 as 2 and 2as3...... \{ } 37 viii

Figure Page

5.9 (3) is obtained by gluing 6 copies of P3, one for each P3.For simplicity,C the indexing elements from 0 for the vertices are2 only shown J for the ﬁrst P3 ( =123).Thecellsthataretobegluedarelabelledby the same color and put in the same position...... 41

5.10 (4) is obtained by gluing 24 copies of P4, one for each P4.For simplicity,C only twelve of the indexing elements from 0 for the2 vertices J are shown for the ﬁrst P4 ( =1234).Onecanﬁndoutwhichcellsare glued...... 42

6.1 Er when n =3...... 48

6.2 Top row: the domains of h = H( , 1); Bottom row: the images of h = H ( , 1)...... · 51 ·

6.3 Ii1 i when n = 6. Notice that Ii1i2i3i4i5i6 = Ii1i2i3 ...... 52 ··· k 3 2 A.1 Deformation retraction from D to S+...... 59 1 2 A.2 Decompositions of P2 and P2 P2 into line segments joined at their ...... ⇥ ⇥ ⇥ 60 C A.3 i, i and , i =2, 3, 5...... 62 N C Si A.4 Two stereographic projections...... 65

3 B.1 The extended closed 3-ball Ext✏(D )...... 66 C.1 The ﬁgure 8 and the ✓ spaces...... 68 C.2 Direct homotopy equivalences between ﬁgure 8 and ✓...... 69 C.3 g f id ; f g id ...... 69 ' 8 ' ✓ C.4 (2)...... 71 F C.5 1-skeleton of (3)...... 71 F C.6 2-cells of (3). Notice that all the 1-cells are drawn in order to show which are the boundariesF of each 2-cell...... 72 C.7 Cact1(2)...... 72 C.8 1-skeleton of Cact1(3)...... 73 C.9 The triangular 2-cells of Cact1(3). More 1-cells are shown in order to illustrate which 1-cells are the boundaries of each 2-cell...... 73 C.10 The rectangular 2-cells of Cact1(3). More 1-cells are shown in order to illustrate which 1-cells are the boundaries of each 2-cell...... 73 C.11 The two generating singular tori of H ( (3))...... 75 2 F ix

Figure Page 1 C.12 The two generating tori of H2(Cact (3))...... 75 1 C.13 The same generators for H2(Cact (3)) with di↵erent topological pictures. 76 C.14 Permutohedral presentation of Cact1(3)...... 76 C.15 Homotopy equivalences between Cact1(3) and (3)...... 77 F C.16 Permutohedron of order 4. It can be realized as the regular octahedron with six pyramids cut o↵...... 78 C.17 Permutohedron of order 4 inside Cact1(4)...... 79 x

SYMBOLS

C2 little 2-cubes operad

C⇤(A, A) Hochschild cochain complex of an associative ring A Cact spineless cacti operad Cact1 normalized spineless cacti quasi-operad (n)permutohedralmodelofCact1(n) C Chen(G)n nth Chen group of the group G k simplex of dimension k

n 1 n 1 D closed unit ball in R

D2 little 2-discs operad E set of white edges of an element in w Tn EndX endomorphism operad for space X F (R2,n)conﬁgurationspaceofn ordered distinct points in R2 F (C,n)sameasabove (n)permutohedralmodelofF (R2,n) F h homotopy equivalence from (n)to (n) n C F H homotopy on permutohedron Pn indexed by

HH⇤(A, A) Hochschild cohomology of an associative ring A ( ,<)posetindexingthecellsof(n) Jn F ( ,<)posetindexingthecellsofP J n i, i subsets of the above indexing i-cells Jn J k initial branching number complete graph operad K [n] 1, 2, ,n { ··· } operad O ⌦2X double loop space xi

⇢ operad morphism

PBn pure braid group on n strands P permutohedron of dimension n 1 n n 2 n 1 S unit sphere in R

Sn symmetric group on [n] scc spineless corolla cacti l = l , , l alistofsequencessuchthat is a shu✏eofthem 1 ··· k 2 3 ··· n S [m , ,m ] the set of all the above l 1 ··· k an element in S , identiﬁed as a total order n 1 2 ··· n S singular chain functor ⇤ ( , )posetindexingthecellsofCact1(n) Tn ! 0 McClure-Smith’s category Tn ( , )posetindexingthecellsofasubdivisionofP T ! n i, i subset sets of the above indexing i-cells Tn T n 1 [l] set of all trees associated to l T n 1 (k)setindexingtreeswithk subtrees over the root white vertex T n 1 set indexing trees with at most k subtrees over the root white T,k vertex

1 1 1 n v the point (( )1, ( )2, , ( )n)inR ··· wi number of edges to the white vertex labelled by i in an element in Tn 1 homotopy equivalence from (n)to (n) n F C xii

ABSTRACT

Zhang, Yongheng PhD, Purdue University, May 2015. Permutohedra, Conﬁguration Spaces and Spineless Cacti. Major Professor: Ralph Kaufmann.

It has been known that the conﬁguration space F (R2,n)ofn distinct ordered points in R2 deformation retracts to a regular CW complex with n! permutohedra

Pn as the top dimensional cells. In this paper, we show that there exists a similar but di↵erent permutohedral structure of the space Cact(n)ofspinelesscactiwithn lobes. Based on these structures, direct homotopy equivalences between F (R2,n)and

Cact(n)arethengiven.Itiswellknownthatthelittle2-discsspaceD2(n)ishomo- topy equivalent to F (R2,n). Our results give partial combinatorial and geometrical interpretation of the equivalences between D2 and Cact. 1

1. INTRODUCTION

1.1 Motivation

Algebraic topology is the study of functors from some category of topological spaces and continuous maps to some category of algebraic objects and algebraic morphisms. One can say that algebraic topology is the study of topological structures using tools from algebra: some topological structures are encoded by algebraic structures. On the other hand, the theory of (topological) operads [29] reverses this pro- cedure: many complicated algebraic structures are encoded by simple topological pictures.

A classic example is the homology of the second loop space of a topological space H (⌦2X)withcoecients in a ﬁeld of characteristic 0. It follows [31] from Fred ⇤ Cohen’s work [10] that H (⌦2X)isaGerstenharberalgebra(amodulewithtwo ⇤ multiplications, one being associative and graded commutative and the other being graded antisymmetric, such that an operation deﬁned by the latter multiplication is aderivationwithrespecttobothmultiplications).Thiscomplicatedalgebraicstruc- ture in homology actually descends from the topological level: it is easy to see that the little 2-cubes operad C2, which has the simple “little cubes inside the unit cube” and “rescale and insert” pictures, acts on ⌦2X.

The above algebraic structure is called Gerstenhaber because Gerstenhaber ﬁrst discovered it in the Hochschild cohomology HH⇤(A, A)ofanassociativeringA [15]. The same algebraic structure present in the above two contexts prompted Pierre Deligne to ask if there is a similar topological explanation of the Gerstenhaber algebra structure on HH⇤(A, A)[31].Inparticular,heaskedifthesingularchainoperad 2

of the little 2-discs D2 (which does not look very di↵erent from C2) or a suitable version of it acts on the Hochschild cochain complex C⇤(A, A)suchthatthisac- ton descends to the Gerstenhaber algebra structure on the Hochschild cohomology

HH⇤(A, A), which was known as Deligne’s (Hochschild cohomology) conjecture.

After initial attempts in [17] and [16], Deligne’s conjecture were proved in [3], [6], [20], [26], [31], [32], [42] and [43], and [44]. While the approach taken by [42] and [43] uses Etingof-Kazhdan quantization and that taken by [44] and [17] uses a filtration of the Fulton-MacPherson compactification of the configuration spaces of points on R2, the other proofs use a version of a topological operad, which was introduced and called by [20] [21] the operad of spineless cacti Cact and it deformation retracts to the quasi-operad of normalized spineless cacti Cact1. There are various generalizations of Deligne’s conjecture (cyclic, A , cyclic-A and higher dimensional versions), leading 1 1 to various new models of Cact as seen in [3], [22], [24], [45], [25], [33] and [23].

The cellular chain operad of spineless cacti CC (Cact), which is isomorphic to ⇤ 1 CC (Cact ), acts on C⇤(A, A)throughthebraceoperad. CC (Cact)isaversion ⇤ ⇤ of the singular chain operad of D2 in the sense that there is a zig-zag of operadic equivalences (being operadic morphisms and homotopy equivalences at the same time) on the topological level connecting D2 and Cact. For example, there are two operadic equivalences pointing in di↵erent directions as follows [20]:

D Cact. 2 • !

This was obtained by noticing that D2(n)andCact(n)arebothK(PBn, 1) spaces where PBn is the pure braid group on n strands and thus the universal covers D2 and Cact] are contractible. In fact, they form the B braid operad [14]. So the morphisms 1 f in D D Cact] Cact] are braid operadic equivalences. After passing to the 2 2 ⇥ ! quotient by pure braid groups PB , we get the above equivalences f f ⇤

D2 = D2/P B (D2 Cact])/P B Cact/P] B = Cact. ⇤ ⇥ ⇤ ! ⇤ f f 3

This method is called the Fiedorowicz’ recognition principle (The B braid op- 1 erad structure of connected Cact guarantees the zig-zag from D2 to Cact). See [13] for an exposition in detail. There is also Berger’s recognition principle which produces zig-zags from C2 and Cact respectively to an operad deﬁned over the complete graph operad [4] [32] [33]. Berger’s method is more combinatorial, yet it heavily uses homotopy colimits, producing big spaces between C2 and Cact. It should also be noted 1 that the homotopy equivalences between D2 and Cact (n)wereﬁrstprovedin[31], where it uses the homotopy invariance of homotopy colimit and the properties of the functors I ,I0 : 0 TOP. The equivalences are given by n n Tn !

colim In hocolim In hocolim hocolim In0 colim In0 , Tn0 Tn0 ! Tn0 ⇤ Tn0 ! Tn0 where is the constant functor. Here, colim In is homotopy equivalent to D2 and ⇤ Tn0 1 colim In0 is homeomorphic to Cact (n). Again, we see a zig-zag of spaces connecting Tn0

D2 and Cact.

Even though there is an zig-zag from D2 to Cact, these two deceptively similar operads have very di↵erent combinatorial, geometrical and operadic structures. It is a mystery even to build a nontrivial direct continuous map from one to the other in terms of pictures (the little discs and cacti conﬁgurations drawn on paper). The original ambitious objective of this project was to construct a direct operadic equivalence from D2 to Cact. This is either extremely dicult or there is an obstruction theory awaiting discovery for proving that this is impossible. The milder objective of this paper is to construct a direct equivariant homotopy equivalence from D2 to Cact with two features:

It does not rely on the K(G, 1) structures that are not present in the little n-disc • operads D for n 3. (c.f. Fiedorowicz’ approach) n It captures the combinatorial and geometrical nature of the two operads but it • does this without going through any bigger space in the middle. (c.f. Berger’s approach) 4

Thereafter, there is still hope that by passing to the chain level, one indeed gets an operadic equivalence S (D2) S (Cact)inthecategoryofdi↵erential graded ⇤ ! ⇤ operads. Combining with the quasi-isomorphism between S (Cact)andCC (Cact), ⇤ ⇤ one would then have an understanding of a direct action of S (D2)onC⇤(A, A). ⇤

1.2 Results

The permutohedra P , n 1playavitalroleinourresults.Tosimplyput,P is n n the convex hull of the point (1, 2, ,n)andtheirpermutationsinRn. ···

Combining a well-known fact and a theorem of [7], we will see that D2(n)deformation retracts (we do not distinguish homeomorphic spaces) to

(n):= Pn/ , F ⇠F an! where glues these n! copies of Pn along their proper faces according to a poset ⇠F . Jn Surprisingly, Cact1 also has a permutohedral structure.

Theorem 5.2.1. For any Sn, colim C is piecewise linearly homeomorphic 2 T (⇠=)toPn.

is a subposet of . consists all trees whose partial orders obtained along T Tn T paths from the root to leaves are compatible with the total order .

It then follows that Cact1(n) is homeomorphic to the space

(n):= Pn/ , C ⇠C an! 5

where glues these n! copies of Pn not only along the proper faces, but also along ⇠C interiors of Pn. Thus, there is a natural projection

1 : (n) (n), n F ! C induced from the identitiy

1 := 1 : P P . n n ! n Sn Sn Sn a2 a2 a2 Theorem 5.4.1. 1 : (n) (n) is a homotopy equivalence. n F ! C

This theorem is proved by constructing a homotopy inverse which is induced by homotopies on these n! copies of P . Composing the homotopy equivalences D (n) n 2 ! (n) 1n (n) Cact1(n) , Cact(n)givesourdesiredmapD Cact. F ! C ⇡ ! 2 !

1.3 Outline

The organization of the paper is as follows. Chapter 2 gives the definition of a topological operad and the definitions of D2 and Cact. Then dimensional analysis are carried out to show why the 3n dimensional D2(n)canbecontractedtoaCW complex of the low dimension n 1, but not a lower dimension. Notice that Cact1(n) is of dimension n 1. Chapter 3 recalls the definition and basic properties of the 2 permutohedron Pn. Chapter 4 reviews the permutohedral structure of F (R ,n). In Chapter 5, we describe the permutohedral structure of Cact(n). The last chapter is on the homotopy equivalence we claimed above. We relegate the proof of two propositions to the appendices where we also document how these results were first discovered. 6

2. THE LITTLE 2-DISCS AND THE SPINELESS CACTI OPERADS

2.1 Operads and their morphisms

Operads exist in many symmetric monoidal categories. For example, there are linear operads (H (D2)), di↵erential graded operads (S (D2)) and topological oper- ⇤ ⇤ ads (D2). We will only consider topological operads starting from this chapter. So by operad, we mean topological operad.

Let us ﬁrst look at an example of an operad which motivates the general deﬁni- tion. For a topological space X and a positive integer n, let EndX (n)bethespace of continuous maps from Xn to X, taking the compact open topology. We consider the collection of these spaces EndX (n) n 1, and call it EndX . { }

Notice that there is a right action of S , the symmetric group on [n]:= 1, 2, ,n , n { ··· } on End (n). To understand this, recall that S acts on Xn from the left: let S X n 2 n and (x , ,x ) Xn, then permutes x , ,x in the sense that it moves x , 1 ··· n 2 1 ··· n i which is in the i’th position in the list to the (i)’th position in the list. This can be written as

(x ,x , ,x )=(x 1 ,x 1 , ,x 1 ). 1 2 ··· n (1) (2) ··· (n)

Given , ⌧ S ,wehave(⌧(x ,x , ,x )) = (x 1 ,x 1 , ,x 1 )= 2 n 1 2 ··· n ⌧ (1) ⌧ (2) ··· ⌧ (n) (x 1 1 ,x 1 1 , ,x 1 1 )=(x 1 ,x 1 , ,x 1 ), which ⌧ ( (1)) ⌧ ( (2)) ··· ⌧ ( (n)) (⌧) (1) (⌧) (2) ··· (⌧) (n) is (⌧)(x ,x , ,x ). So we do see that S acts on Xn from the left. Now let 1 2 ··· n n f End (n)and S . We deﬁne the action of on f by (f )(x ,x , ,x )= 2 X 2 n · 1 2 ··· n f((x ,x , ,x )). It follows that S acts on End (n)fromtheright. 1 2 ··· n n X 7

Elements from these EndX (n)spacescanbecomposedandtherearemanyways of doing so because they are multi-input functions. Let f End (m), g End (n) 2 X 2 X and i be any integer from 1 to m. Then we deﬁne the element f g End (m+n 1) i 2 X by

(f ig)(x1,x2, ,xm+n 1)=f(x1,x2, ,xi 1,g(xi,xi+1, ,xi+n 1),xi+n, ,xm+n 1). ··· ··· ··· ···

It follows that these structures have at least three properties:

(Equivariance) Let f End (m), g End (n), S , ⌧ S and i = • 2 X 2 X 2 m 2 n 1, 2, ,m, then (f ) (g ⌧)=(f g) ( ⌧), where ( ⌧)isan ··· · i · (i) · i i element in Sm+n 1 which permutes the m entries in the list

(x1,x2, ,xi 1, (xi, ,xi+n 1),xi+n, ,xm+n 1) ··· ··· ···

according to ((xi, ,xi+n 1)isasingleentryinthisstep)andthenthe ··· entries in (xi, ,xi+n 1)arepermutedaccordingto⌧. ··· (Associativity) Let f End (l), g End (m), h End (n)andi = • 2 X 2 X 2 X 1, 2, ,l, j =1, 2, ,l + m 1. Then two ways of composing the three ··· ··· elements yield the same element:

(f j h) i+j 1 g 1 j i 1   8 (f i g) j h = f i (g j i+1 h) i j i + m 1 >   <> (f j i+1 h) i gi+ m j l + m 1.   > > (Indentity) The identity: function 1 End (1) is special: for any f End (n), • 2 X 2 X 1 f = f and f 1=f where i =1, 2, ,n. 1 i ···

The collection of spaces EndX = EndX (n) n 1 with the above structures form an { } operad. In general, we have the following definition axiomatizing the above structures possessed by EndX . This definition can be found, for example, in [28]. It is equivalent to the original definition in [29]. 8

Deﬁnition 2.1.1 A sequence of spaces = (n) n 1 form an operad if for any n, O {O } S acts on (n) and for all m, n and i =1, 2, ,m, there is a continuous map n O ···

: (m) (n) (m + n 1) i O ⇥ O ! O

(a, b) a b, 7! i called the operadic composition, such that the following conditions hold.

(Equivariance) Let a (m), b (n), S , ⌧ S and i =1, 2, ,m, • 2 O 2 O 2 m 2 n ··· then (a ) (b ⌧)=(a b) ( ⌧). · i · (i) · i

(Associativity) Let a (l), b (m), c (n) and i =1, 2, ,l, j = • 2 O 2 O 2 O ··· 1, 2, ,l+ m 1. Then ···

(a j b) i+j 1 c 1 j i 1   8 (a i b) j c = a i (b j i+1 c) i j i + m 1 >   <> (a j i+1 b) i ci+ m j l + m 1.   > > (Indentity (optional)) There: is an element 1 (1) such that for any a (n), • 2 O 2 O 1 a = a and a 1=a where i =1, 2, ,n. 1 i ···

Deﬁnition 2.1.2 Given two operads and , an operadic morphism ⇢ : O P O ! P is a collection of continuous maps ⇢ : (n) (n), preserving all the operadic n O ! P structures, i.e.,

(operadic composition) For any m, n, a (m), b (n) and i =1, 2, ,m, • 2 O 2 O ··· ⇢m+n 1(a i b)=⇢m(a) i ⇢n(b). (equivariance) For any n, a (n) and S , ⇢ (a )=⇢ (a) . • 2 O 2 n n · n ·

(identity (optional)) ⇢1(1 )=1 , where 1 is the identity of and 1 is the • O P O O P identity of . P 9

Example. There is an operadic morphism ⇢ : D End where X is a based 2 ! X 2 2 2 second loop space ⌦ Y . In this case, we say D2 acts on ⌦ X and ⌦ X is an algebra over D2.

Deﬁnition 2.1.3 An operadic morphism ⇢ : is called an operadic equivalence O ! P if each ⇢ : (n) (n) is a homotopy equivalence. n O ! P

Deﬁnition 2.1.4 If there is a zig-zag of operadic equivalences connecting to the O little 2-discs operad D (or the little 2-cubes operad C ), then is called an E -operad. 2 2 O 2

Example. We saw in Chapter 1 that the spineless cacti operad Cact is an E2- operad. Next, we recall the deﬁnitions of D2 and Cact.

Remark. In the deﬁnition of operads, we require the identity to exist. Sometimes, the existence of identity is absent. These were called pseudo-operads in [28]. But we don’t make the distinction here. We call both operads and pseudo-operads simply operads.

2.1.1 The little 2-discs operad D2

For any n 1, the little 2-discs operad D has underlying spaces 2 D (n)= (x , ,x ,r , ,r ) (D˚2)n (0, 1]n : r + r x x for all i = j 2 { 1 ··· n 1 ··· n 2 ⇥ i j  | i j| 6 r 1 x for all k . k  | k| }

Each element in D2(n)parametrizesaconﬁgurationofn little 2-discs inside the unit disc such that the interiors of the little discs do not intersect and the interior of each little disc is contained in the unit disc by specifying the centers xi and radii ri of the little discs. S acts on D (n)by(x , ,x ,r , ,r ) =(x , ,x ,r , ,r ). n 2 1 ··· n 1 ··· n · (1) ··· (n) (1) ··· (n) It has a pictorial representation as follows. 10

Figure 2.1. A little 2-discs conﬁguration determined by an element in D2(5).

Figure 2.2. Action of S5 on D2(5).

Let a =(x , ,x ,r , ,r ) D (m) and b =(y , ,y ,s , ,s ) 1 ··· m 1 ··· m 2 2 1 ··· n 1 ··· n 2 D (n), then a b deﬁned as the following element in D (m + n 1) 2 i 2

(x1, ,xi 1,xi+riy1, ,xi+riyn,xi+1, ,xm,r1, ,ri 1,ris1, ,risn,ri+1, ,rm). ··· ··· ··· ··· ··· ···

Pictorially, a b is obtained by rescaling the entire conﬁguration represented by b i to the size of the little disc labelled by i in a and then inserting it into the ith disc in a to get a conﬁguration of n + m 1littlediscsinsidetheunitdisk.Thenthelabels 1, 2, ,n of the little discs in b are changed to i, i +1, ,i+ n 1andthelabels ··· ··· i +1,i+2, ,mof the little discs in a are changed to i + n, i + n +1, ,m+ n 1. ··· ··· 11

Figure 2.3. An example of operadic insertion 2 : D2(3) D2(2) D2(4). The dashed circle is not part of the picture. ⇥ !

The identity is (x ,r ) D (1) where x =0andr =1. 1 1 2 2 0

One can check that the equivariance, associativity and identity conditions are satisﬁed. So D2 is an operad.

2.1.2 The spineless cacti operad Cact

The operad of spineless cacti Cact was introduced in [20]. We ﬁrst brieﬂy review

Cact = Cact(n) n 1 as topological spaces here. We refer the reader to the original { } article for the details.

1 2 1 2 1. An Sr in the plane R is an orientation preserving embedding Sr R where ! 1 2 Sr is the standard circle of radius r in R . We also call these embeddings parametrizations, and each image point corresponds to a unique parameter. The image of (r, 0) is called the local zero.

1 n 2. Given a conﬁguration of nSr s, i =1, ,n,inR , labelled by 1, 2, ,n, i ··· ··· whose number of intersection points is ﬁnite, we construct its dual black and white graph on the plane by replacing each circle with a white vertex with labels 12

ri and i, and each intersection of circles a black vertex followed by connecting each white vertex with each black vertex on the circle (which the white vertex represents) by an edge. We will only consider conﬁgurations of circles in R2 whose dual graph is a planar tree from now on. A planar tree has a cyclic order for each set of the edges adjacent to each vertex.

3. A conﬁguration of circles in the plane is called rooted if one of the circles is marked by a point, which is called the global zero. Correspondingly, in the dual black and white graph construction, we include a special black vertex (called the root vertex) and connect it to the adjacent white vertex to make the dual tree also rooted. A rooted planar tree also has a linear order for each set of incoming edges to a vertex except the root vertex.

4. By specifying on which component the root lies, we also make a planar rooted tree planted. Such trees also has a linear order for the set of edges incident to the root vertex.

5. A conﬁguration of circles in the plane is called tree-like if its dual graph is a connected planar tree and none of the circles is contained in another.

6. As a set, the spineless cacti Cact(n)equalsthesetofallrootedplantedtree-like conﬁgurations of nS1 s, i =1, ,n, labelled 1, 2, ,nin the plane such that ri ··· ··· the local zeros are all at the intersections and the local zero of the root lobe coincides with the global zero, modulo isotopies preserving the parameters for the intersections. 13

7. As a set, the normalized spineless cacti Cact1(n)isdeﬁnedthesameway,except that r , i =1, ,n, are required to be 1. Correspondingly, the dual black and i ··· white trees only have 1, 2, ,n as labels. ···

8. The topology of Cact1(n)isthatofaregularCWcomplex.Foreachdualblack

and white tree ⌧ above, let Ew be the set of edges which point from a black vertex to a white vertex and we call them the white edges. We also let v be | i| v the number of incoming edges of the white vertex labelled i. Let | i| be the

vi +1 ˚ vi geometric simplex (x0, ,xvi ) R| | xi 0,x0 + + x vi =1 and | | { ··· | | 2 | ··· | | } n ˚ ˚ vi its interior. The open cell C⌧ indexed by this tree ⌧ is deﬁned as i=1 | |.

n vi Then the closure of the cell indexed by this tree C⌧ equals i=1 Q| |. Each

number xi of a simplex represents an arc on a lobe (circle). TheQ attaching map is deﬁned by setting one number from one simplex by 0, which gives the projec-

tion from @C⌧ to the union of the closure of the cells indexed by the degenerate trees whose corresponding arcs are contracted. One can readily verify that this CW complex as a set is bijective to Cact1(n).

1 n 9. The topology of Cact(n)isdeﬁnedastheproducttopologyofCact (n) R>0. ⇥ One can also verify that the latter set is bijective to Cact(n). So Cact(n)deformation retracts to Cact1(n).

An element of Cact(5) is given below. Notice that for any c Cact(n), if one 2 start from the root vertex (the black square) and travel around the perimeter of the conﬁguration then one will eventually come back to the root vertex. The path trav- elled is called the outside circle.

As in D , S acts on Cact(n)bypermutingthelabels. : Cact(m) Cact(n) 2 n i ⇥ ! Cact(m + n 1) is deﬁned similarly: Given c Cact(m)andc Cact(n), c c is 1 2 2 2 1 i 2 14

Figure 2.4. An element in Cact(5): it is the isotopy class of orientation and intersection parameter preserving embeddings of the ﬁve standard 37 3 circles of radii 2, 3, 30 , 1, 2 such that the images form a rooted planted tree-like conﬁguration of circles. The local zeros are denoted by black dots. The global zero is denoted by a black square.

obtained by rescaling the outside circle of c2 to that of the i’th circle of c1 and then identifying the outside circle of the resultant conﬁguration to the i’th lobe of c1.

Figure 2.5. An example of operadic insertion 2 : Cact(3) Cact(2) Cact(4). ⇥ !

The element 1 Cact(1) having radius 1 is only a right-handed identity: for any 2 c Cact(n)andi =1, ,n, c 1=c. But 1 c = c does not hold. 2 ··· i 1 15

One can check that the above structures make Cact an operad (more precisely, pseudo-operad).

2.2 Dimensions

Recall that in order to construct a homotopy equivalence from D2 to Cact,we 1 ﬁrst want to ﬁnd a equivalence from D2 to Cact . Notice that D2(n)isaspaceof dimension 3n and Cact1(n) is a space of dimension of n 1. In this section, we would like to analyze the lowest possible dimension of the homotopy type of D2 and look for evidence that D Cact1 should exist. 2 !

If X is a topological space, then the conﬁguration space of n distinct ordered points in X is deﬁned as follows:

F (X, n)= (x , ,x ) Xn x = x for any 0 i

Because the Euclidean plane R2 is the underlying space of C, we also denote F (R2,n)byF (C,n). This is not only for notational convenience, but also for some other reasons which we will see later. Let us study the topological type of F (C,n). 16

1. F (C, 1) = C = R2, which deformation retracts to a point.

2. F (C, 2) C C⇤, where C⇤ is C with the origin removed. This fact uses the ⇡ ⇥ property that (C, +) is a topological group: the homeomorphism is given by 1 (z1,z2) (z1,z2 z1). F (C, 2) deformation retracts to S . 7!

3. F (C, 2) C C⇤ C⇤ 1 . This also uses the fact that C⇤ under complex ⇡ ⇥ ⇥ \{ } multiplication is a topological group: the homeomorphism is given by the com-

z3 z1 position (z ,z ,z ) (z ,z z ,z z ) (z ,z z , )(seethediscussion 1 2 3 1 2 1 3 1 1 2 1 z2 z1 7! 7! on page 23 of [11]). F (C, 3) deformation retracts to S1 (S1 S1). ⇥ _

4. If we can ﬁnd a group structure on C⇤ 1 , then we would be able to present \{ } F (C, 4) as a trivial bundle over F (C, 3). However, this is impossible (see page 81-83 of [12] for a proof).

In general, for n 4,

1 1 1 n 1 1 F (C,n) S (S S ) ( S ). 6' ⇥ _ ⇥ ···⇥ _i=1

This can be proved by the following method in [9]. The fundamental group of F (C,n) 1 1 1 n 1 1 is PB ,thepurebraidgrouponn strands, while that of S (S S ) ( S ) n ⇥ _ ⇥···⇥ _i=1 is F1 F2 Fn 1, where Fi is the free group on i generators. Let G be any ··· group, then the Chen groups Chenn(G)ofG are the successive quotients of the lower central series of H := G/[[G, G], [G, G]], i.e.,

Chen1 = H/[H, H]

Chen2 =[H, H]/[[H, H],H]

Chen3 =[[H, H],H]/[[[H, H],H],H]

··· 17

p Theorem 2.2.1 (Cohen-Suciu [9]) Let q =0if p

Table 2.1. Ranks of Chen(PBn)k and Chen(F1 Fn 1)k, where k occupies the top row. ···

123 4 5 6 ··· PB n n 2 n+1 3 n+1 4 n+1 5 n+1 n 2 3 4 4 4 4 ··· n n n+1 n+2 n+3 n+4 F1 Fn 1 2 3 4 5 ··· 2 3 4 5 6 7 ···

Thus, Chen(PBn)k =Chen(F1 Fn 1)k when n, k 4. Since the Chen groups 6⇠ ··· are invariants of group isomorphic types, it follows that PBn = F1 Fn 1 when 6⇠ ··· n 4.

We observed that for n =1, 2, 3, F (C,n)deformationretractstoaCWcomplex of dimension n 1. These were derived from their homotopy types as products of wedges of circles. We saw that F (C,n) does not have the same homotopy type as a product of wedges of circles for n 4, but we will argue that F (C,n)deformation retracts to a CW complex of dimension n 1foralln.

Theorem 2.2.2 (Andreotti-Frankel [1]) If M is a Stein manifold with dimCM = m, then M has the homotopy type of a CW complex with dim M m. R  Stein manifold has two equivalent deﬁnitions, one in the language of several complex variables and the other in the language of di↵erential topology. Here, a Stein manifold is a complex manifold which can be biholomorphically embedded as a closed subset in CN for some N. The proof uses Morse theory. One can refer to [1] or to Chapter 6 and Chapter 7 of the notes by Milnor [35] for the proof. 18

F (C,n)isacomplexmanifoldwithdimC = n. It deformation retracts to a submanifold with dim = n 1: C

n r1(F (C,n)) = (z1, ,zn) C z1 + + zn =0,zi = zj for any 0 i

As a motivational example in the real case, consider the submanifold M := x { 2 2 R x =0 of R, which is not closed in R. But M 0 := (x, y) R xy =1 , as a space 6 } { 2 } homeomorphic to M, is closed in 2. R

Under the map (z1, ,zn) (z0,z1, ,zn 1), r1(F (C,n)) is homeomorphic to ··· 7! ···

n F (n):= (z0,z1, ,zn 1) C P (z0,z1, ,zn 1)=1 , { ··· 2 | ··· } where P is a polynomial deﬁned by

n 1 P (z0,z1, ,zn 1)=z0 (zi zj) (z1 + + zn 1 + zk). ··· ··· 1 i

i Definition 2.2.1 Given a space X, let b , i 0, be its Betti numbers. Then 1 b t i i=0 i is called the Poincaréseries of X.IfX has the homotopy type of a CW complexP of i dimension N, then bi =0for i>N and i1=0 bit is called the Poincarépolynomial of X. P

Theorem 2.2.3 (Arnold [2]) The Poincar´epolynomial of F (C,n) is (1 + t)(1 + 2t) (1 + (n 1)t). Thus, F (C,n) has the homotopy type of a CW complex with ··· dimension at least n 1.

We sketch the proof here. The original proof uses cohomology. Let us use homology (with integer coecients). Let pn : E = F (C,n) B = F (C,n 1) be the projec- ! tion map forgetting the last point zn, i.e., pn :(z1, ,zn) (z1, ,zn 1). It can be ··· 7! ··· shown that this is a locally trivial fibration with fiber C having n 1pointsremoved, which deformation retracts to a wedge of n 1circles.Thefundamentalgroupofthe base ⇡1(B)isPBn 1 which does not permute the set of the n 1 removed points. Thus, the action of ⇡1(B)onthehomologyofthefiberistrivial.SoontheE2 page of the 2 n 1 1 n 1 1 Serre spectral sequence, each term E = H (B; H ( S )) = H (B) H ( S ). p,q p q _i=1 ⇠ p ⌦ q _i=1 Thus, the E2 page has only two nonzero rows.

2 2 2 2 2 1 E0,1 E1,1 E2,1 E3,1 E4,1 d2 d2 d2 d2

0 H0(B) H1(B) H2(B) H3(B) H4(B)

0 1 2 3 4 20

The ﬁbration pn admits a section sn : B E deﬁned by (z1, ,zn 1) ! ··· 7! (z1, ,zn 1,zn), where ···

z2 = z1 +1 ifn=2, z1 + + zn 1 zn = ··· +max1 i2. n 1   | | So (pn) is surjective. It then follows from the Gysin’s sequence that the di↵erentials ⇤ d2 are zeros. 0

2 d 2 Er 1,1 Er1 1,1 0

Hr(E) 0 (pn) ⇤ 2 d 2 0 Er,10 Hr(B) Er 2,1 Er1 2,1 0

0 Hr 1(E) (pn) ⇤ d2 0 Er1 1,0 Hr 1(B)

n 1 1 So E = = E2. Therefore, H (F (C,n)) = H (F (C,n 1)) H ( i=1 S ). By 1 ··· ⇤ ⇠ ⇤ ⌦ ⇤ _ induction,

1 1 1 n 1 1 H (F (C,n)) = H (S ) H (S S ) H ( i=1 S ). ⇤ ⇠ ⇤ ⌦ ⇤ _ ⌦ ···⌦ ⇤ _

The assertion follows.

So F (C,n), and thus D2(n), has the homotopy type of a CW complex with the smallest dimension n 1. In Chapter 4, we will see that F (C,n)deformationre- tracts to an (n 1)-dimensional regular CW complex obtained by gluing n! copies of permutohedra Pn. Before that, let us review the deﬁnition of Pn in the next chapter. 21

3. PERMUTOHEDRA

In this chapter, we recall the deﬁnition of permutohedra. They play vital roles in our results.

3.1 Permutohedra as convex hulls

For n 1, recall that S means the group of permutations on [n]:= 1, 2, ,n . n { ··· } Let S . To save space, we denote the image of i under not by (i), but 2 n by . We identify with the sequence of its images . For example, i 1 2 ··· n =3142 S means :1 3, 2 1, 3 4and4 2. So 1 2 3 4 2 4 7! 7! 7! 7! 1 1 1 1 1 =( ) ( ) ( ) ( ) =2413 S . Note that each S determines 1 2 3 4 2 4 2 n atotalorderon[n].

n 1 1 1 Given Sn, we define the vector v in R as (( )1, ( )2, , ( )n). So if 2 ··· 4 =3142,thenv =(2, 4, 1, 3) R . As the reader will find out later, it is necessary 2 1 to define v using instead of letting v be ( , , , ). 1 2 ··· n

Deﬁnition 3.1.1 The permutohedron P is the convex hull of the set of points v n { 2 n R Sn . So | 2 } n Pn = tv R t =1,t 0 . { 2 | } Sn Sn X2 X2

Figure 3.1 illustrates the ﬁrst three permutohedra Pn, n =1, 2, 3.

Two vertices v , v of P , n 2areadjacentifandonlyifv is obtained from ⌧ n ⌧ v by switching two coordinate values di↵ering by 1 so that the Euclidean distance from v to v⌧ is the minimal number p2. For example, in P3,(2, 1, 3) is adjacent to 22

Figure 3.1. The permutohedra P1, P2 and P3. Each vertex is labelled on the left and v on the right.

(3, 1, 2), and to obtain (3, 1, 2) from (2, 1, 3), we swith 2 with 3. Correspondingly, we switch the numbers in consecutive positions of to get ⌧. For instance, we switch the numbers in the second and third positions of 213 to get 231, which correspond to (2, 1, 3) and (3, 1, 2), respectively.

The following facts can also be readily checked.

n 1. The vertex set of Pn is v R Sn . { 2 | 2 }

n (n+1)n 2. Pn is contained in the hyperplane (x1, ,xn) R x1 + + xn = . { ··· 2 | ··· 2 }

3. P is a polytope of dimension n 1. n 23

3.2 Permutohedra as realizations of posets

Deﬁnition 3.2.1 For any S , the poset consists of all sequences of numbers 2 n J with bars a a a , k 1, where each a =(a ) (a ) is a nonempty sub- 1| 2|···| k i i 1 ··· i mi sequence of such that k (a ) , , (a ) =[n]. (We say a a is a 1 ··· n i=1{ i 1 ··· i mi } 1 ··· k rearrangement of 1 n.) The` partial order < is generated by removing a bar and ··· merging the numbers as follows

a1 ai 1 ai ai+1 ai+2 ak < a1 ai 1 b ai+2 ak, (?) ···| | | | |···| |···| | | |···| where b = b b is a rearrangement of a a . If an element in contains 1 ··· mi+mi+1 i i+1 J i bars, then we deﬁne its order to be n 1 i. Let i, i =0, 1, ,n 1, be the set J ··· consisting of elements of order i in . J

Remark. In the above deﬁnition, by the deﬁnition of the set , b must be a J subsequence of . It follows that a and a are subsequences of b.Sowe 1 ··· n i i+1 have an equivalent characterization of b:

b is a subsequence of . • 1 2 ··· n b is a shu✏eofa and a (This means b is the union of the two disjoint • i i+1 subsequences ai and ai+1).

Example. For =145372896 S ,wehave15349 76 28 < 153 4796 28, which 2 9 | | | | | are elements in 5 and 6, respectively. J J

Deﬁnition 3.2.2 Let be the realization functor from to the category of topo- F J logical spaces deﬁned by

1 2 n = v F | |···| n 0 on order 0 elements and a to be the convex hull of b R b , b a for F {F 2 | 2 J  } general a . So = P . 2 J F n 24

Fix i. a i is exactly the collection of i dimensional faces of P and if {Fa| 2 J} n a = a1 a2 ak where ai =(ai)1 (ai)m , then a1 a2 a is anely isomorphic to | |···| ··· i F | |···| k P P P . We refer the reader to [4], [8], [19], [35], [28] and [46] for m1 ⇥ m2 ⇥ ···⇥ mk more details. Finally, we deﬁne on < to be face inclusions. The example of P for F 4 =1234 S is given in Figure 3.2. 2 4

Figure 3.2. The codimension 1 faces of P4 and their indexing elements in 2 . Visible faces are labelled by bold-faced numbers. The faces of the J1234 types abc d, ab cd and a bcd are anely isomorphic to P3 P1, P2 P2 and P |P , respectively.| | ⇥ ⇥ 1 ⇥ 3 25

4. PERMUTOHEDRAL STRUCTURE OF F (R2,n)

2 F (R ,n)deformationretractstoaspacewhichisobtainedbygluingn! copies of Pn. We ﬁrst describe the gluing data through a poset which contains all the n! posets Jn introduced in the previous chapter. J

4.1 The poset Jn

Definition 4.1.1 As a set, the poset n equals the union Sn . The partial J 2 J order of n is defined the same way as that in (?) except b isS defined di↵erently: J

1. We still require b to be a shu✏e of ai and ai+1.

2. We don’t require b to be a subsequence of for a particular S . 1 2 ··· n 2 n Example. 1234 is the only element in that is greater than 13 24. But in J1234 | Jn where n =4,theelementsgreaterthan1324 are 1324, 1234, 1243, 2134, 2143 and | 2413.

Deﬁnition 4.1.2 We extend from , S to naturally and let F J 2 n Jn

(n):=colim n . F J F Remark. So (n) is obtained by gluing n! copies of P along their proper faces F n according to the partially order set . Alternatively, we can write Jn

(n)= Pn / , F ⇠F Sn ! a2 where for x Pn indexed by and y Pn indexed by ⌧, x y if there are 2 2 ⇠F a such that x and y have the same coordinates in (we simply write 2 J \ J⌧ Fa x = y in the future). 2 Fa 26

4.2 A theorem of Blagojevi´cand Ziegler

Theorem 4.2.1 ( [7]) (n) is homeomorphic to a deformation retract of F (R2,n). F

Remark. In fact, [7] described regular CW complex models which are homeomorphic to deformation retracts of the conﬁguration spaces F (Rk,n)forallk, n 1, which were used in their proof when n is a prime power of the conjecture of Nan- dakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter. The same CW complex models were also studied in [4] and [17] and they were called the Milgram’s permutohedral model in [4].

We briefly review the proof of the above theorem here. First, R2n deformation 2 retracts to the subspace Wn in which the geometric center of each configuration is 2 n shifted to the origin. We denote this retraction by r . Then W 0 is partitioned 1 n \ into relatively open infinite polyhedral cones. These cones give the Fox-Neuwirth

2 n stratiﬁcation of W 0 and they constitute a partially ordered set. Next, a relative n \ interior point for each cone is chosen. These points yield the vertices of a star-

2 n shaped PL ball. Then W 0 radially deformation retracts to the boundary of n \ this PL ball. We denote this retraction by r2. Finally, the Poincar´e-Alexander dual 2 2 n 2 complex of r2 r1(F (R ,n)) relative to r2 (Wn 0 r1(F (R ,n))) is constructed, which \ \ 2 is a deformation retract of r2 r1(F (R ,n)). Let this third retraction be r3. In 2 2 conclusion, F (R ,n)deformationretractstor3 r2 r1(F (R ,n)), which has a partially ordered set structure with the partial order the reverse of that of the Fox-Neuwirth stratiﬁcation. This partially ordered set is precisely and (n)ishomeomorphic Jn F 2 to r3 r2 r1(F (R ,n)). 27

5. PERMUTOHEDRAL STRUCTURE OF Cact1(n)

5.1 Reformulation of Cact1(n) as the realization of a poset

Let ( , )denotethepartiallyorderedsetofrootedplanarplantedbipartite Tn ! (black and white) trees with white leaves, a black root, and n white vertices labelled from 1 to n, where is generated by identifying two adjacent edges connected to a ! white vertex (contraction of an arc in the cacti picture).

Figure 5.1. Example of four elements of and their morphisms . Tn !

Let i be the subset of such that E = i (Recall that E is the set of white Tn Tn | w| w edges, i.e., the set of edges which point from a black vertex to a white vertex). So 0 Tn n 1 0 consists of the minimal elements in and the maximal elements. is also Tn Tn Tn the set of trees indexing the spineless corolla cacti SCC(n)[20].Weletscc()bethe following element in 0. Tn 28

Then recall that as a set,

1 Cact (n)= C˚⌧ .

⌧ n a2T As a space, 1 Cact (n)= C⌧ / , n 1 ⇠ ⌧ 2aTn where x y for x C and y C if there is ⌧ with ⌧ ⌧ , ⌧ such that x ⇠ 2 ⌧1 2 ⌧2 2 Tn 1 2 and y have the same coordinates in C (We simply write x = y C in the future). ⌧ 2 ⌧

Let C be the realization functor from the poset ( , )tothecategoryoftopo- Tn ! logical spaces such that

1. For ⌧ , C = w1 w2 wn , where w is the number of incoming 2 Tn ⌧ ⇥ ⇥ ···⇥ i edges to the white vertex labelled by i.

2. If in ⌧ 0 ⌧, ⌧ 0 is obtained from ⌧ by identifying the jth and the (j +1)th ! incoming edges of the white vertex i (where we deﬁne the 0th and the (wi +1)th incoming edges to be the outgoing edge of this white vertex), then we deﬁne

C( ) = idw1 id wi 1 @j id wi+1 idwn , ! ⇥ ···⇥ ⇥ ⇥ ⇥ ···⇥

where

w 1 w @ : i i j ! (t0,t1, ,tw 1) (t0,t1, ,tj 1, 0,tj, ,tw 1). ··· i 7! ··· ··· i

Then it follows that 1 Cact (n)=colim n C. T 29

Figure 5.2. A subposet of and its image under the functor C. Tn

1 5.2 Permutohedra Pn in Cact (n).

To describe the permutohedral structure of Cact1(n), for any S , we introduce 2 n the subset ( , )of( , )asfollows. T ! Tn !

Deﬁnition 5.2.1 The elements of are the trees in such that for any leaf vertex, T Tn the sequence of labels on the white vertices along the shortest path from the root vertex to this leaf vertex is a subsequence of . Some examples of trees in are 1 ··· n T53214 given below. The partial order of is the restriction of that of . T Tn

Remark. We say that consists of all trees in whose partial order is com- T Tn patible with the total order .

The above deﬁnition extends from automorphisms on [n] to bijections from [n]to any set of distinct positive integers with cardinality n. We denote such functions by 30

Figure 5.3. Examples of trees in and the associated cacti pictures. T53214

BJ . If BJ , then as in the case for S , we identify with the sequence of its n 2 n n images . 1 2 ··· n

Deﬁnition 5.2.2 Let BJ . Let 1 be the identity in S . Then the set con- 2 n n n T sists of all trees obtained by replacing each label i of a white vertex on a tree in T1n by (i).If⌧ and ⌧ are trees in , then we denote the associated trees in by 1 2 T1n T (⌧ ) and (⌧ ). The partial order of is then deﬁned as follows: (⌧ ) (⌧ ) in 1 2 T 1 2 whenever ⌧ ⌧ in . T 1 2 T1n

Similarly, we have the sets i, i =0, 1, ,n 1 of trees of i white edges. T ···

Example. If : 1, 2, 3 2, 5, 7 maps 1 5, 2 7and3 2, then BJ { } ! { } 7! 7! 7! 2 3 and we have 31

Figure 5.4. The sets 2 , 1 , 0 . T572 T572 T572

n 1 Let S . We would like to partition and then ﬁlter , the set of maximal 2 n T elements in according to the number of branches on top of the unique white vertex T adjacent to the root black vertex (which we call the root white vertex). We let k be this number. So k can take values from 1 to n 1. We call k the initial branching number.

Deﬁnition 5.2.3 The following is the most important deﬁnition in this paper.

Let S . Let k, 1 k n 1, be the above initial branching number and • 2 n   m ,m , ,m be k positive integers such that m + + m = n 1. Each 1 2 ··· k 1 ··· k mi denotes the number of white vertices of the i’th subtree we graft to the root white vertex.

Let l := (l ) (l ) (l ) , i =1, ,k be subsequences of such that the dis- • i i 1 i 2 ··· i mi ··· joint union of the underlying sets k (l ) , (l ) , , (l ) is , , , . i=1{ i 1 i 2 ··· i mi } { 2 3 ··· n} Notice that the missing of 1 is not` a mistake.

We denote each l , l , , l by l and the set of all such l by S [m , ,m ]. • 1 2 ··· k 1 ··· k

n 1 n 1 We deﬁne [l] to be the set of all trees ⌧ (⌧ , ⌧ , , ⌧ ) in obtained • T 0 1 2 ··· k T by grafting ⌧ , , ⌧ to ⌧ , where ⌧ is the only element in , and ⌧ is an 1 ··· k 0 0 T1 i m 1 element in i . The following picture illustrates what we mean by “grafting”. Tli 32

Figure 5.5. An example of grafting of trees and the associated cacti picture.

For a ﬁxed initial branching number k, we deﬁne •

n 1 n 1 (k)= [l]. T T m1, ,mk l S[m1, ,m ] a··· 2 a··· k n 1 Elements in (k) represent all cacti conﬁgurations with k cacti sub-conﬁgurations T on top of the root lobe.

Then we have n 1 • n 1 n 1 = (k), T T ka=1 i.e., the set of maximal trees in is partitioned by the initial branching number. T Then we define • n 1 n 1 = (q). T,k T q k a So n 1 n 1 n 1 n 1 n 1 n 1 (1) = ,1 ,2 ,n 2 ,n 1 = T T ⇢ T ⇢ ···⇢ T ⇢ T T n 1 gives a filtration of . T Let [l] be the subset of such that each element in [l] is less than an • T T T n 1 element in [l]. Similarly, we define . We also have the inherited poset T T,k structures on [l] and . T T,k

Example. The elements of 5 [l , l ] where l =36,l =214areshownin T532146 1 2 1 2 Figure 5.6. 33

Figure 5.6. The elements of 5 [l] where l = l , l and l =36,l =214. T532146 1 2 1 2

The realization functor C on restricts to ( , ), ( [l], )and( , ), Tn T ! T ! T,k ! respectively.

1 We can ﬁnd n! permutohedra Pn in Cact (n)asshownbelow.

Theorem 5.2.1 For any Sn, colim C is piecewise linearly homeomorphic (=) 2 T ⇠ to Pn.

Proof We proceed by induction. When n =1, 2, colim C are a point and a closed T line segment, respectively. So the statement is true in these two cases. Suppose the statement is true for all m and all S where m

Now, let Sn. We will ﬁrst show that colim C is a PL (piecewise linear) ball 2 T n 1 of dimension n 1. (We simply say colim C is a D .) T

By the induction hypothesis and the deﬁnition of the realization functor C, for l S [m , ,m ], 2 1 ··· k k colim [l]C = Pm1 Pm2 Pm . T ⇠ ⇥ ⇥ ···⇥ k ⇥

n 1 So colim [l]C is a D . We will iteratively use the following fact: T

n 1 n 2 n 2 If X and Y are both D and i : D , X and j : D , Y are injective ! ! n 2 n 2 maps such that i(D )istheunionofsomefacetsofX and j(D )istheunion 34

n 2 n 2 n 2 of some facets of Y (so both i(D )andj(D )areD ), then the pushout of

n 2 n 1 X - D , Y is a D . !

1 When k =1,weknowl = 2 n and colim ,1 C =colim [l]C = Pn 1 is ··· T T ⇠ ⇥ n 1 a D .

n 1 Now suppose for 2 k n 1, colim ,k 1 C is a D .Foranym1,m2, ,mk   T ··· with m 1andm + + m = n 1, and then any l S [m , ,m ], i 1 ··· k 2 1 ··· k n 1 n 1 colim [l]C,asaD , is glued to the D : colim ,k 1 C along Pm1 Pmk T T ⇥ ··· ⇥ ( k v v v ), where v v = k and v v v corresponds to the i=2 1 ··· i ··· k+1 1 ··· k+1 1 ··· i ··· k+1 k k+1 contractionS of the i’th arc on the root lobe. Notice that since @ = i=1 v1 vi vk+1 b b ··· ··· k 1 k 1 k k 1 is a S and v2v3 vk+1 v1v2 vk is a D , i=2 v1 vi Svk+1 is a D . ··· ··· ··· ··· b k n 2 Thus, Pm1 Pmk ( i=2Sv1 vi vk+1)isaDS . So the resultant space X0 ⇥ ··· ⇥ ··· ··· b n 1 is a D . Then, take anotherS l S[m1, ,mk]. Again, colim [l]C is glued to 2b ··· T n 2 k X0 along a D : P P ( v v v ), so the resultant space is a m1 ⇥ ··· mk ⇥ i=2 1 ··· i ··· k+1 n 1 D . We glue these colim [l]C to colimS ,k 1 C one after another until we exhaust l T T b and m , ,m . Then we have 1 ··· k

colim C is a PL ball of dimension n 1. T,k

Therefore, colim C =colim,n 1 C is a PL ball of dimension n 1. Indeed, we T T have the following ﬁltration of the PL ball colim ,n 1 C by PL balls: T

colim ,1 C colim ,2 C colim ,n 2 C colim ,n 1 C. T ⇢ T ⇢ ···⇢ T ⇢ T 35

Figure 5.7. colim C, i =1, 2, 3. T1234,i

Next, we show that the PL ball colim C is indeed piecewise linearly isomorphic T to P . Let us deﬁne a new functor. For any = S , let be the realization n 1 ··· n 2 n C functor from to the category of topological spaces deﬁned by T 1 1 1 n scc() =(( )1, ( )2, , ( )n) R C ··· 2 0 on order 0 elements and to be the convex hull of ⌧ 0 , ⌧ 0 ⌧ for general C⌧ {C⌧ 0 | 2 T  } ⌧ . Let the image of under be face inclusions. 2 T ! C

Each cell on the boundary of colim C is indexed by a tree obtained by concate- T m 1 nating ⌧ , ⌧ , , ⌧ at the root vertex, where ⌧ i and a =(a ) (a ) is 1 2 ··· k i 2 Tai i i 1 ··· i mi a subsequence of such that a a is a rearrangement of .Wecall 1 ··· n 1 ··· k 1 ··· n such a tree ⌧ ⌧ . An example is given below. 1|···| k

Let = ⌧ ⌧ ⌧ : ⌧ . We shall glue the cells indexed Ta1 |Ta2 |···|Tak { 1| 2|···| k i 2 Tai } m1 1 m2 1 mk 1 by ⌧1 ⌧2 ⌧k together. By the induction hypothesis | |···| 2 Ta1 |Ta2 |···|Tak and the piecewise linear homeomorphism from colim a C to colim a by extend- T i T i C ing the vertex correspondences C , we know for each a a a , scc() 7! Cscc() 1| 2|···| k 36

colim a a a = Pm1 Pm2 Pm , which is the characterization of the T 1 |T 2 |···|T k C ⇥ ⇥ ··· ⇥ k cells of Pn.

Therefore, colim = Pn and thus colim C = Pn. T C T ⇠

Remark. The above result gives n!/2decompositionsofPn into products of simplices. It is likely that they had been know in [5]. See Remark 1.10 in the aforementioned article.

5.3 Permutohedral structure in Cact1(n) in other contexts

The above permutohedral structure also appears when we see Cact1 as a cellular

E -preoperad [4] or as a colimit over the poset category 0 [31]. First, let us recall 2 T n the deﬁnition of preoperad and the complete graph preoperad from [4] below.

⇤ is the category whose objects are the ﬁnite nonempty sets [n] and whose morphisms are the injective maps. A preoperad is a contravariant functor from ⇤ to a O category. We denote the image of [n]by and that of ⇤(m, n)by⇤ : . On 2 On ! Om

Notice that any injective map in ⇤(m, n)canbefactoredasabijectivemapin ⇤(m, m)followedbyanincreasingmapin⇤(m, n). If ⇤(m, n)and S ,we 2 2 n let ⇤()betheinjectivepartof . If ⇤(2,n)and(1) = i, (2) = j, then n 2 (2) is also denoted ij. Let N be the set of all edge-labelings of the complete graph n (2) on n vertices by positive integers. An element of N is denoted µ =(µij)1 i (j). 37

n (2) The complete graph preoperad : ⇤ Poset is deﬁned by n = N Sn on K ! K ⇥ objects and for ⇤(m, n), ⇤(µ, ) (⇤(µ), ⇤()). The partial order on is 2 7! Kn deﬁned by

(µ, ) (⌫, ⌧) for all i

1 Proof Let = Cact (n), and if ⇤(m, n)andc , we let ⇤(c) be the cacti On 2 2 On conﬁguration obtained by contracting the lobes of c with labels in [n] (1), , (m) \{ ··· } and then changing label (i)toi, i =1, ,m. It can be checked that is functorial. ··· O

Figure 5.8. Let c be an element in Cact1(5) shown on the left. Let ⇤(3, 5) be deﬁned by (1) = 5, (2) = 1 and (3) = 2. Then to get 2 ⇤(c), we ﬁrst contract lobes labelled by [5] 5, 1, 2 and then relabel 5 as 1, 1 as 2 and 2 as 3. \{ }

There is a natural ﬁltration of by (k) = (µ, ) µ k for i

Let us recall the deﬁnition of cellular decomposition of a space by a paritally ordered set from [4]. Let A be a partially ordered set and X atopologicalspace.

Then a collection (c↵)↵ A of closed contractible subspaces of X is called a cellular 2 A-decomposition of X if the following three conditions hold: 38

1. c c ↵ ; ↵ ⇢ , 

2. the cell inclusions are closed ﬁbrations;

3. X =colimAc↵.

Notice that our = Cact1(2) admits a cellular (2)-decomposition (↵), ↵ (2), O2 K2 O2 2 K2 compatible with the S2 action, which is given by

(2,12) (2,21) (1,12) (1,21) = C⌧1(⌧2), = C⌧2(⌧1), = C⌧1 ⌧2 , and = C⌧2 ⌧1 , O O O | O | where ⌧ is the only element in and ⌧ is the only element in . 1 T1 2 T2 (2) Then for any n 3and↵ n , we have the formally deﬁned cell 2 K

(↵) 1 ij⇤ (↵) = (⇤ ) ( ). On ij O2 1 i

Corollary 5.3.0.1 Let Sn. Let µ =(µij)1 i

1 ij⇤ (µ,) 1 Proof (⇤ ) ( )consistsofallelementsinCact (n)suchthat ij O2

(1) if (i) < (j)andifthelabelsi and j are on the same path from the root lobe to a leaf lobe, then i precedes j;

(2) if (i) > (j)andifthelabelsi and j are on the same path from the root lobe to a leaf lobe, then j precedes i. 39

1 ij⇤ (µ,) 1 Therefore, 1 i

Next, we review the partially ordered set 0 of [31]. T n

Let a and b be two (partially or totally) ordered sets. By a b, we mean a is ✓ contained in b where a and b are viewed as sets of all ordered pairs i

The objects of 0 are pairs (t, p), where t is a total order and p is a partial order, T n both of [n], subjecting to the following condition:

p t,andifi

p p and t top p . 1 ✓ 2 2 \ 1 ✓ 2 It can be readily checked that the poset of trees ( , )isisomorphictotheposet Tn ! of formulas of [31]. Given a formula f , we have a total order t of [n]as In 2 In f the sequence of numbers when we read f from left to right, and a partial order pf of [n] generated by the pairs i

i( j ) . Then (t ,p )isanobjectin 0 . ··· ··· ··· ··· f f T n 40

1 Let In0 (t, p)bethefollowingsubcomplexofCact (n):

In0 (t, p)= Cf . f (t ,p ) (t,p) { | f [f  } Let S . Notice that the sequence = determines a total order of [n] 2 n 1 ··· n and that a total order is a special partial order. We have the following result.

Corollary 5.3.0.2 In0 (, ) ⇠= Pn.

Proof Notice that f (t ,p ) (, ) = = . Again, the result follows from { | f f  } I ⇠ T Theorem 5.2.1.

5.4 The spaces (n) C

Deﬁnition 5.4.1 We extend from to and then let C T Tn

(n):=colim n . C T C

Immediately, we have L : (n) Cact1(n). n C ⇡

Alternatively, by Theorem 5.2.1 we can write

(n)= Pn / , C ⇠C Sn ! a2 where we understand that P indexed by takes the subdivision by and for x P n T 2 n indexed by and y Pn indexed by ⌫, x y if there is ⌧ ⌫ such that x = y 2 ⇠C 2 T \ T in . Examples when n =3andn = 4 are shown in Figure 5.9 and Figure 5.10, C⌧ respectively.

The two spaces (n)and (n)arecloselyrelated: F C 41

Figure 5.9. (3) is obtained by gluing 6 copies of P3, one for each P3. For simplicity,C the indexing elements from 0 for the vertices are2 only J shown for the ﬁrst P3 ( =123).Thecellsthataretobegluedare labelled by the same color and put in the same position.

1. They are both obtained by gluing n! copies of Pn.

2. But the gluings for (n)onlyoccurontheproperfacesofP while those for F n (n)alsohappenintheinteriorofP . In fact, only the interiors of the hyper- C n

cubes ⌧1(⌧2( (⌧n) )) in each of the n! copies of Pn, where ⌧i ,arenotglued. C ··· ··· 2 T i

3. If x y where x Pn indexed by and y Pn indexed by ⌫, let a = a1 ak ⇠F 2 2 |···| be the element in such that a and x = y in the interior of . Jn 2 J \ J⌫ Fa Then we can ﬁnd ⌧ = ⌧ ⌧ where ⌧ , i =1, ,k, by using 1|···| k 2 T \ T⌫ i 2 Tai ··· cacti subdivisions of each Pm such that x = y in ⌧ .Sox y. Thus, there is i C ⇠C anaturalprojection 1 : (n) (n), n F ! C 42

Figure 5.10. (4) is obtained by gluing 24 copies of P4, one for each P . For simplicity,C only twelve of the indexing elements from 0 for 2 4 J the vertices are shown for the ﬁrst P4 ( =1234).Onecanﬁndoutwhich cells are glued. 43 induced from the identitiy

1 := 1 : P P . n n ! n Sn Sn Sn a2 a2 a2

We will show in the next section that 1n is a homotopy equivalence by constructing an inverse homotopy equivalence. Let us record the above fact here:

Theorem 5.4.1 1 : (n) (n) is a homotopy equivalence. n F ! C 44

6. HOMOTOPY EQUIVALENCES BETWEEN (n) and F (n) C In this section, we will describe inverse homotopy equivalences h : (n) (n) n C ! F h h P P , h induced from n := Sn : Sn n Sn n where each has to map 2 2 ! 2 all points in Pn other` than those` in the interior` of ⌧1(⌧2( (⌧n) )), where ⌧i i ,to C ··· ··· 2 T proper faces of Pn in order to have hn(x) hn(y)ifx y. ⇠F ⇠C

We will ﬁnd each h by ﬁrst describing homotopies H : P I P satisfying n ⇥ ! n the following conditions and then letting h = H ( , 1) for each . ·

( 1) H ( , 0) = 1 : P P . ⇤ · n ! n

( 2) If x y where x is in Pn indexed by and y is in Pn indexed by ⌫, then ⇤ ⇠F H(x, t) H⌫(y, t)forallt I. ⇠F 2

( 3) If x y where x is in Pn indexed by and y is in Pn indexed by ⌫, then ⇤ ⇠C ( 3a) H(x, t) H⌫(y, t)forallt I,and ⇤ ⇠C 2 ( 3b) H(x, 1) H⌫(y, 1). ⇤ ⇠F

So h 1 : (n) (n) and 1 h : (n) (n)arebothinducedfrom n n F ! F n n C ! C hn : Sn Pn Sn Pn, and we have 1 (n) H hn 1n and 1 (n) H 1n hn 2 ! 2 F ' F C ' C where`H and H `are both induced from Sn H. Before describing the homo- F C 2 topies H, we introduce two constructions:` extended products of permutohedra and a simplex, and iterated cones, which will play vital roles in the construction of H. 45

6.1 Extended products of permutohedra and a simplex

Let n 3. For m , ,m 1, where k 2andm + + m = n 1, we 1 ··· k 1 ··· k mi 1 k k view each Pmi as a subspace of R and asubspaceofR . Let us consider the product of k permutohedra and the simplex k: P P P k in m1 ⇥ m2 ⇥ ···⇥ mk ⇥ m1 1 m2 1 m 1 k R R R k R , which we saw is homeomorphic to the closed unit ⇥ ⇥ ···⇥ ⇥ n 1 n 1 k ball D in R , with the boundary @(Pm1 Pm2 Pmk ), i.e., the following ⇥ ⇥···⇥ ⇥ union (@P P P k) (P @P P k) (P m1 ⇥ m2 ⇥ ···⇥ mk ⇥ m1 ⇥ m2 ⇥ ···⇥ mk ⇥ ··· m1 ⇥ k n 2 Pm2 Pmk @ ), being homeomorphicS to the unit sphere S S. S ⇥ ···⇥ ⇥

Let v ,v , ,v be the vertices of k.Weknowthat 1 2 ··· k+1 k+1 P P P @k = P P P ( v v v ), m1 ⇥ m2 ⇥ ···⇥ mk ⇥ m1 ⇥ m2 ⇥ ···⇥ mk ⇥ 1 ··· i ··· k+1 i=1 [ b which contains the subspace P P P ( k v v v ). m1 ⇥ m2 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 S b Notice that, being the union of two simplices along v v v , v v v v v v 2 3 ··· k 1 2 ··· k 2 ··· k k+1 k 1 k k is homeomorphic to D .So v1 vi vk+1 = @ Int(v1v2 vk Sv2 vkvk+1) i=2 ··· ··· \ ··· ··· k 1 k is also homeomorphic to D S. Hence, Pm1 Pm2 Pmk ( i=2 v1 S vi vk+1) b ⇥ ⇥···⇥ ⇥ ··· ··· n 2 k is homeomorphic to D and so is @(Pm1 Pm2 Pmk S ) Int(Pm1 Pm2 ⇥ ⇥ ···⇥ ⇥ \ b ⇥ ⇥ P ( k v v v )). If we decompose P P P k ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 m1 ⇥ m2 ⇥ ···⇥ mk ⇥ k as a fiber spaceS over Pm1 Pm2 Pmk ( i=2 v1 vi vk+1)whosefibersare b⇥ ⇥ ···⇥ ⇥ ··· ··· k singletons along @(Pm1 Pm2 Pmk (S i=2 v1 vi vk+1)) and the fibers ⇥ ⇥ ···⇥ ⇥ ···b ··· over the points not in the previous set are closedS intervals, then we can contract b P P P k onto @(P P P k) Int(P P m1 ⇥ m2 ⇥ ···⇥ mk ⇥ m1 ⇥ m2 ⇥ ···⇥ mk ⇥ \ m1 ⇥ m2 ⇥ P ( k v v v )). In fact, we have the following result, whose proof ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 is relegated toS the appendix. b Proposition 6.1.1 @(P P P k) Int(P P P m1 ⇥ m2 ⇥ ···⇥ mk ⇥ \ m1 ⇥ m2 ⇥ ···⇥ mk ⇥ ( k v v v )) is a deformation retract of P P P k. i=2 1 ··· i ··· k+1 m1 ⇥ m2 ⇥ ···⇥ mk ⇥ S b 46

Let (P P ( k v v v )) I be the embedded image into m1 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k ⇥ ✏ n 1 k R of a trivial ﬁber bundleS over Pm1 Pmk ( i=2 v1 vi vk+1)with b⇥ ··· ⇥ ⇥ ··· ··· k ﬁber the interval I✏ := [0, ✏], whose intersection with PSm1 Pmk is ⇥ ··· ⇥b ⇥ P P ( k v v v ). m1 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 S b Let Ext (P P k)betheunionofP P k and ✏ m1 ⇥ ··· ⇥ mk ⇥ m1 ⇥ ··· ⇥ mk ⇥ (P P ( k v v v )) I . We also relegate the proof of the m1 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 ⇥ ✏ following propositionS to the appendix. b

k Proposition 6.1.2 There is a homotopy HExt✏(Pm Pm ) : Ext✏(Pm1 1 ⇥···⇥ k ⇥ ⇥ ··· ⇥ P k) I Ext (P P k), satisfying the following conditions: mk ⇥ ⇥ ! ✏ m1 ⇥ ···⇥ mk ⇥

1. it contracts P P P k onto @(P P P m1 ⇥ m2 ⇥ ··· ⇥ mk ⇥ m1 ⇥ m2 ⇥ ··· ⇥ mk ⇥ k) Int(P P P ( k v v v )); \ m1 ⇥ m2 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 S b 2. it maps (P P ( k v v v )) I homeomorphically to m1 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 ⇥ ✏ k Ext✏(Pm1 Pmk ); S ⇥ ···⇥ ⇥ b

3. Let X be space and I0 a closed subinterval of I, then we call a map G :

X I0 X the identity homotopy on X if G( ,t)=1 for all t I0. Then ⇥ ! · X 2 k k HExt✏(Pm Pm ) is the identity homotopy on @Ext✏(Pm1 Pmk ). 1 ⇥···⇥ k ⇥ ⇥ ···⇥ ⇥

Now we describe how to construct the extended products inside each Pn.

Let n 3. We ﬁrst consider the identity 1 =12 n S .Let2 k n 1. n ··· 2 n   For any m , ,m 1withm + + m = n 1, we consider the special 1 ··· k 1 ··· k j = j , , j S [m , ,m ] where j =23 (m +1)and j =(m + + 1 ··· k 2 1n 1 ··· k 1 ··· 1 i 1 ··· mi 1 +2) (m1 + + mi +1)fori =2, ,k. Notice that the string of numbers ··· ··· ··· 47

k 1j1 jk is the string 12 n for 1n. We know that colim 1 [j] = Pm1 Pm . ··· ··· T n C ⇠ ⇥···⇥ k ⇥ k Let Extm , ,m (colim 1n [j] )betheimageinPn of Ext✏(Pm1 Pmk ) under 1 ··· k T C ⇥ ···⇥ ⇥ an extended homeomorphism m1, ,m whose property we specify later. ··· k

Then for general S , let r = r , , r S [m , ,m ]. Let ! = r r 2 n 1 ··· k 2 1 ··· k 1 1 ··· k 2

Sn. We let Extm , ,m (colim [r] )betheimageofExtm , ,m (colim 1n [j] )under 1 ··· k T C 1 ··· k T C the linear map

t⌫ scc(⌫1, ,⌫n) t⌫ scc((!⌫)1, ,(!⌫)n). C ··· 7! C ··· ⌫ Sn ⌫ Sn X2 X2

Since for fixed and k, any two from the collection of colim [r] for all m1, ,mk T C ··· and all r S [m , ,m ] either are disjoint or share a subspace homeomorphic to 2 1 ··· k i D , where i is at most n 3, let us choose the homeomorphisms m1, ,m such that ··· k the interiors of Extm , ,m (colim [r] )arepairwisedisjointforfixed and k. Since 1 ··· k T C for fixed n and the homeomorphisms m1, ,mk , Extm , ,m (colim [r] )onlydepends ··· 1 ··· k T C on r, for simplicity, we denote it by Er.

Under the homeomorphisms m1, ,m and the linear maps, we transfer the homo- ··· k k k topies HExt✏(Pm Pm ) from Ext✏(Pm1 Pmk )toeachEr. We call this 1 ⇥···⇥ k ⇥ ⇥ ···⇥ ⇥ homotopy H : E I E . It has all the three properties as those in Proposition r r ⇥ ! r 6.1.2.

Corollary 6.1.0.1 The homotopies H : E I E satisfy the following conditions: r r⇥ ! r

1. Choose S such that r is compatible with it, i.e., any r =(r ) (r ) of 2 n i i 1 ··· i mi r = r , , r , is a subsequence of . For each i =1, 2, ,k 1, let 1 ··· k 2 ··· n ···

n 2 m 1 d [r]= ⌧ (⌧ , , ⌧ ⌧ , , ⌧ ) ⌧ , ⌧ j iT { 0 1 ··· i| i+1 ··· k | 0 2 T1 j 2 Trj } 48

and k 1 n 2 n 2 d [r]= d [r]. T iT i=1 [ Let d [r] be the set of trees in such that each tree is smaller than an element T T d n 2 H @ in [r]. Then r contracts colim [r] onto ( colim [r] ) Int(colimd [r] ); T T C T C \ T C

2. it maps the closure of Er colim [r] homeomorphically to Er; \ T

3. Hr is the identity homotopy on @Er.

Figure 6.1. Er when n =3. 49

6.2 Iterated cone construction

Let n 4and1n =12 n Sn. Notice that colim 1 ,1 , as a subspace of ··· 2 T n C =colim = P , is the cartesian product of P with I , where I 1n 1n n n 1 pn(n 1) pn(n 1) F T C is the interval [0, n(n 1)]. We construct the iterated cones of the faces of P of n codimension 1, wherep P is seen as the realization of 1 under , as follows. n n F

Step 1. For any dimension 2 face a1 an 2 of Pn, let v be a point in the interior F |···| of Pn with distance ✏2 directly below the geometric center of a1 an 2 , where by F |···|

”directly below” we mean the line joining v with the geometric center of a1 an 2 F |···| is perpendicular to a1 an 2 . We also require that v is not in colim 1n,1 . Then we F |···| T C form the join a1 an 2 v, which is a cone with base a1 an 2 . We call such a cone F |···| ⇤ F |···| 1 C ( a1 an 2 ), which is 3-dimensional. F |···|

Step i 1, 3 i n 3. For any dimension i face a1 an i of Pn, we consider   F |···| its union with the (i 2)-cones of its codimension 1 faces:

i 2 a1 an i C ( b). |···| F [ i 1 F b ,b

Let v be a point in the interior of Pn with distance ✏i directly below the geometric center of a1 an i . Then we form the join of v with this union and denote it by F |···| i 1 C ( a1 an i ). We call it the (i 1)-cone of a1 an i , which is (i+1)-dimensional. F |···| F |···|

Step n 3. For any dimension n 2(codimension1)face a1 a2 of Pn, we form F | the union n 4 B(a1 a2):= a1 a2 C ( b). | | F [ n 3 F b ,b

For the special elements 1 2 n and 2 n 1, we let Cyl( 1 2 n)=Cyl( 2 n 1) | ··· ··· | F | ··· F ··· | be the space as the union of line segments such that each line segment joins a point in B(1 2 n) to the corresponding point in B(2 n 1). We call it the cylinder of | ··· ··· | 1 2 n (or of 2 n 1). F | ··· F ··· | 50

For general a a (including 1 2 n and 2 n 1), let v be a point in the interior 1| 2 | ··· ··· | of Pn with distance ✏n 2 directly below the geometric center of a1 a2 such that v is F | not in the interior of Cyl( 1 2 n). We form the join B(a1 a2) v and denote it by F | ··· | ⇤ n 3 C ( a1 a2 ). We call it the (n 3)-cone of a1 a2 . F | F |

We choose the values of ✏2, , ✏n 2 small enough and the orientation of each v ··· such that

1. all the previous conditions are satisﬁed;

2. each iterated cone is contained in the union of the elements of a collection {Cal } n 1 such that each a has the largest possible arity number; l 2 T1n 3. the union of all (n 3)-cones exhibits maximal symmetry.

To construct the iterated cones for P as for any S , we take as images of n F 2 n the interated cones of under the linear map F1n

t⌫ ⌫1 ⌫n t⌫ (⌫)1 (⌫)n . F |···| 7! F |···| ⌫ Sn ⌫ Sn X2 X2 We will use the following many times.

Let P be a polytope and H : P I P ahomotopywithH ( ,t) =1 P ⇥ ! P · |@P @P for all t I. A cone P v can be identiﬁed with P I/P 1 .WedeﬁneHP I : 2 ⇤ ⇥ ⇥ { } ⇥ (P I) I P I by ((x, s),t) (HP (x, t),s). So HP I ( , ,t) (@P ) I =1(@P ) I ⇥ ⇥ ! ⇥ 7! ⇥ · · | ⇥ ⇥ for all t I. HP I induces a homotopy on P I/P 1 P v. We call it HP v. 2 ⇥ ⇥ ⇥ { } ⇡ ⇤ ⇤ Let us record the following fact.

Lemma 6.2.1 HP v satisﬁes HP v( ,t) (@P ) v =1(@P ) v. ⇤ ⇤ · | ⇤ ⇤ 51

6.3 The homotopy

Now we describe the homotopies H where S for some n 1. 2 n

For n =1, 2, we let H be the identity homotopies. In fact, (n) (n)inthese C ⇡ F two cases. For n =3,weletH ( ,t)beH ( ,t)onE and H ( ,t)onE ; · 2,3 · 2,3 3,2 · 3,2 we let H be the identity homotopy on P Int(E E ). By Corollary 6.1.0.1, 3\ 2,3 [ 3,2 H is a well-deﬁned homotopy on P3. See Figure 6.2.

Figure 6.2. Top row: the domains of h = H( , 1); Bottom row: the images of h = H ( , 1). · ·

For each n 4, we describe H in n steps, assuming we know the homotopies for all m

Step 1: t [0, 1 ]. We need the following deﬁnitions. 2 n

Let f m : I I be deﬁned by t 1 t and f m : I I by t 1 (t + m 1), where 0 ! 7! m 1 ! 7! m m 4. Let f m : I I and f m : I I be the identity maps if m =3, 2, 1. Then 0 ! 1 ! n n 1 n k+1 we let fi1i2 ik = fi fi fi for 1 k n, where ij =0, 1, j =1, ,n. ··· 1 2 ··· k   ···

We also let Ii1i2 i = fi1i2 i (I). For a = a1 ak , let ai be the length of the ··· k ··· k |···| 2 J string a and a the maximum of a i =1, ,k . i { i ··· } 52

Figure 6.3. Ii1 i when n = 6. Notice that Ii1i2i3i4i5i6 = Ii1i2i3 . ··· k

From now on, let i1 =0.

Now For a = a1 an 2 ,ifa =2,thenallbuttwoofa1, , an 2 are of |···| 2 J ··· 1 1 1 1 length 1 and we let HC ( a) : C ( a) [0, ] C ( a)betheidentityhomotopy;if F F ⇥ n ! F a =3,thena = a where a = ijk and all the other a sareoflength1.We ···| l|··· l m 1 deﬁne H a : a [0, ] a by F F ⇥ n ! F

1 1 (x, t) a H 123 ( al (x),fi1 in 3 (t)) 7! ⇤⇥···⇥⇤⇥ F ··· ⇥⇤⇥···⇥⇤ ⇣ ⌘ if t Ii1i2 in 3 for some i2, ,in 3 and (t, x) (t, x)otherwise,where a = 2 ··· ··· 7! ( , , , , , , ): is the homeomor- ⇤ ··· ⇤ al ⇤ ··· ⇤ Fa !⇤⇥···⇥⇤⇥F123 ⇥⇤⇥···⇥⇤ phism deﬁned under i 1, j 2andk 3. Then we get the induced homotopy 7! 7! 7! 1 1 1 1 HC ( a) : C ( a) [0, ] C ( a). F F ⇥ n ! F

Let 3 i n 3. Suppose we have described the homotopies for t [0, 1 ]on   2 n 1 the (i 2)-cones. For a = a1 an i, deﬁne H a : a [0, ] a by |···| F F ⇥ n ! F

1 1 1 (x, t) a H 12 a ( a1 (x),fi1i2 in a (t)), ,H 12 a ( an i (x),fi1i2 in a (t)) , ··· 1 1 ··· n i n i 7! F ··· ··· F ··· ⇣ ⌘ where a =( a1 , , an i ): a 12 a1 12 an i is the homeomorphism ··· F ! F ··· ⇥ ···⇥ F ··· under the assignments (a ) k, for k =1, 2, ,a and j =1, 2, ,n i,andwe j k 7! ··· j ··· 1 1 deﬁne H ( (x),f (t)) to be (x)iff (t)= . The homotopies 12 aj aj i1i2 in a aj i1 in a F ··· ··· j ··· j ; i 1 H a and HCi 1( ), where b and b < a, agree on their overlaps. F Fb 2 J 53

i 2 Thus, we get a well-deﬁned homotopy on i 1 C ( ), a1 an i b ,b

1 Lastly, for a = a1 a2, deﬁne H a : a [0, ] a by | F F ⇥ n ! F 1 1 1 (x, t) H ( (x),f (t)),H ( (x),f (t)) a 12 a1 a1 i1i2 in a 12 a2 a2 i1i2 in a 7! F ··· ··· 1 F ··· ··· 2 ⇣ ⌘ as above. Again, we get well-deﬁned homotopy on

n 4 B(a1 a2)= a1 a2 C ( b). | | F [ n 3 F b ,b

Cyl( 1 2 n) B(1 2 n) I. Then we also have the homotopy HCyl( 1 2 n) : F | ··· ! | ··· ⇥ F | ··· 1 1 Cyl( 1 2 n) [0, ] Cyl( 1 2 n)deﬁnedby(x, t) ✓ HB(1 2 n)(✓1(x),t), ✓2(x) . F | ··· ⇥ n ! F | ··· 7! | ··· n 3 The homotopies HCyl( 1 2 n) and the HC ( a)sagreeontheiroverlaps.Ontheother F | ··· F hand, we let H : P [0, 1 ] P be the identity homotopy on the complement n ⇥ n ! n n 3 n 2 of the interior of Cyl( 1 2 n) a 1 2 n,2 n 1 C ( a). By Lemma 6.2.1, F | ··· [ 2J \{ | ··· ··· | } F 1 H : Pn [0, ] Pn is a well-deﬁnedS homotopy. ⇥ n !

j 1 j Step j, j =2, ,n 1: t [ , ]. Let H ( ,t)beH ( , nt j +1)onE , ··· 2 n n · r · r where r = r , , r is compatible with (recall this means each r is a subsequence 1 ··· j i of ), and let H ( ,t)betheidentityhomotopyonthecomplementoftheinte- 2 ··· n · j 1 j rior of the union of these E sinP . By Corollary 6.1.0.1, each H : P [ , ] P r n n ⇥ n n ! n is a well-deﬁned homotopy.

n 1 n 3 n 3 Step n: t [ , 1]. We get the induced homotopies HC ( a) : C ( a) 2 n F F ⇥ n 1 n 3 n 2 [ , 1] C ( )foralla as those in the Step 1 except that we let i =1 n ! Fa 2 J 1 1 n 1 and we don’t consider the cylinders. On the other hand, we let H : P [ , ] P n⇥ n n ! n n 3 n 2 be the identity homotopy on the complement of the interior of a C ( a). By 2J F n 1 Lemma 6.2.1, H : Pn [ , 1] Pn is a well-deﬁned homotopy.S ⇥ n ! 54

The above n homotopies agree on their overlaps, thus we get a well-deﬁned homotopy H : P I P . n ⇥ ! n

Theorem 6.3.1 For any n 1, h : (n) (n) is a homotopy equivalence. n C ! F

Proof There is nothing to check for n =1, 2.

By our previous discussion, it suces to prove that the homotopies H , S 2 n satisfy ( 1), ( 2) and ( 3). ⇤ ⇤ ⇤

As a warm-up, let us consider the case for n =3indetailﬁrst.(1) holds by ⇤ Corollary 6.1.0.1. Let x y where x Pn indexed by and y Pn indexed by ⇠F 2 2 ⌫. Then x = y in Int( )forsomea . If the order of a is 2, then = ⌫. Fa 2 J \ J⌫ So H (x, t)=H (x, t)=H (y, t)in . Otherwise (the order of a is smaller than ⌫ ⌫ Fa 2), H (x, t)=x = y = H (y, t)in because we have the identity homotopy on the ⌫ F↵ 1-skeleton. So H(x, t) H⌫(y, t)foranyt, verifying ( 2). Let x y where x Pn ⇠F ⇤ ⇠C 2 indexed by and y P indexed by ⌫. Then there is a 2 2 with the lowest 2 n 2 T \ T⌫ arity number such that x = y in . Then for any t, either H (x, t)=H (y, t)in Ca ⌫ Ca or H (x, t)=H (y, t)in where b 2 2 and b has arity number one greater ⌫ Cb 2 T \ T⌫ than or equal to that of a. Thus, ( 3a)holds.Lastly,ifx = y in Int( )wherea ⇤ Ca has arity number 1, then = ⌫ and so H (x, 1) = H (y, 1) = H (y, 1) in = ; ⌫ F F⌫ otherwise, H (x, 1) = H (y, 1) in for some b 1 1. Hence, ( 3b)holds. ⌫ Fb 2 J \ J⌫ ⇤

For n 4, from the description of H when t [0, 1 ], we see that ( 1) holds. 2 n ⇤ Now let x y where x Pn indexed by and y Pn indexed by ⌫. Then x = y in ⇠F 2 2 Int( )forsomea . Then H (x, t)=H (y, t)in (but not necessarily in Fa 2 J \ J⌫ ⌫ Fa Int( a)). Thus, ( 2) holds. Now let x y where x Pn indexed by and y Pn F ⇤ ⇠C 2 2 n 1 n 1 indexed by ⌫. Then there is a with the lowest arity number such that 2 T \ T⌫ x = y in . Then for any t, either H (x, t)=H (y, t)in or H (x, t)=H (y, t) Ca ⌫ Ca ⌫ n 1 n 1 in where b and b has arity number greater than or equal to that Cb 2 T \ T⌫ 55

of a. Therefore, ( 3a) holds. Finally, let x y where x Pn indexed by and ⇤ ⇠C 2 y P indexed by ⌫, so there is a such that x = y in . Then there 2 n 2 T \ T⌫ Ca n 1 n 1 is b = b b such that H ( , )=H ( , ) . By the 1|···| k 2 F \ F⌫ Ca n ⌫ Ca n ⇢ Fb deﬁnition of the homotopy, there is b0 < b such that H (x, 1) = H (y, 1) in , ⌫ Fb0 establishing ( 3b). ⇤ REFERENCES 56

REFERENCES

[1] A. Andreotti, T. Frankel, The Lefshetz theorem on hyperplane sections, Ann. Math. 69 (1959), pp. 713-717. [2] V. I. Arnold, The cohomology ring of the colored braid group,originallypubl. in Mat. Zametki 5 (1969), pp. 227-231. Translated by V. Vassiliev (2014). [3] M. A. Batanin, C. Berger, The lattice path operad and Hochschild cochains, Comtemp. Math. 504 (2009), pp. 23-52.

[4] C. Berger, Combinatorial models of real conﬁguration spaces and En-operads, Contemp. Math. 202 (1997), pp. 37-52.

[5] C. Berger, Cellular structures for En-operads, Workshop on Operads, Os- nabr¨uck, (1998), pp. 4-22. [6] C. Berger, B. Fresse, Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc. 137 (2004), pp. 135-174. [7] P. V. M. Blagojevi´c, G. M. Ziegler, Convex equipartitions via equivariant obstruction theory, Israel J. Math 200 (2014), pp. 49-77. [8] G. Carlsson, J. Milgram, Stable homotopy and iterated loop spaces, Handbook of Algebraic Topology, North-Holland (1995), pp. 505-583. [9] D. C. Cohen, A. I. Suciu, The Chen groups of the pure braid group, Comtemp. Math. 181 (1995), pp. 45-64.

[10] F. R. Cohen, The homology of Cn+1-spaces, n 0, Homology of Iterated Loop Spaces, Lecture Notes in Mathematics, Spring-Verlag (1976), no. 533, pp. 207- 351. [11] F. R. Cohen, Configuration spaces, their homology and Lie algebras, Journal of Pure and Applied Algebra, 100 (1995), pp. 19-42. [12] E. R. Fadell, S. Y. Husseini, Geometry and Topology of Configuration Spaces, Springer Monographs in Mathematics, 2001. [13] B. Fresse, Homotopy of operads and Grothendieck-Techmüller groups,bookin preparation, http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/

[14] Z. Fiedorowicz, Construction of En operads, Workshop on Operads, Osnabr¨uck, (1998), pp. 34-55. [15] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. 78 (1963), no. 2, pp. 267-288. 57

[16] M. Gerstenhaber, A. A. Voronov, Homotopy G-algebras and moduli space operad, Intern. Math. Res. Notices (1995), no. 3, pp. 141-153. [17] E. Getzler, J. Jones, Operads, homotopy algebras and iterated integrals for double loop spaces, Preprint hep-th/9409063 [18] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. [19] M. M. Kapranov, Permuto-associahedron, MacLane coherence theorem and asymptotic zones for the KZ equation, J. Pure and Applied Algebra 85 (1993), pp. 119-142. [20] R. M. Kaufmann, On several varieties of cacti and their relations, Algebraic & Geometric Topology 5 (2005), pp. 237-300. [21] R. M. Kaufmann, On spineless cacti, Deligne’s conjecture and Connes- Kreimer’s Hopf algebra, Topology 46 (2007) pp. 39-88. [22] R. M. Kaufmann, A proof of a cyclic version of Deligne’s conjecture via cacti, Math. Res. Lett. 15 (2008), no. 5, pp. 901-921. [23] R. M. Kaufmann, Dimension vs. Genus: A surface realization of the little k- cubes and an E -operad, in: Algebraic Topology - Old and New. M. M. Post- nikov Memorial1 Conference, Banach Center Publ. 85 (2009), Polish Acad. Sci., Warsaw, pp. 241-274. [24] R. M. Kaufman, R. Schwell, Associahedra, cyclohedra and a topological solution to the A Deligne conjecture, Adv. Math. 223 (2010), no. 6, pp. 2166-2199. 1 [25] M. Kontsevich, Operds and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, pp. 35-72. [26] M. Kontsevich, Y. Soibelman, Deformations of algebras over operads and the Deligne conjecture, In Conférence MoshéFlato 1999, Vol. I (Dijon), volume 21 of Math. Phys. Stud., pp. 255?307. Kluwer Acad. Publ., Dordrecht, 2000. [27] M. Markl, Cohomology operations and the Deligne conjecture, Czechoslovak Math. J. 57 (132) (2007), pp. 473-503. [28] M. Markl, S. Shnider and J. Stashe↵, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96., American Mathematical Society, Providence, RI, 2002. [29] J. P. May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271 (1972). [30] W. S. Massey, The homotopy type of certain configuration spaces, Bol. Soc. Mat. Mex. Vol. 37 (1992), pp. 355-365. [31] J. E. McClure, J. H. Smith, A solution of Deligne’s Hochschild cohomology conjecture, in Recent Progress in Homotopy Theory (Baltimore, MD, 2000), Contemp. Math. 293 (2002), pp. 153-193. [32] J. E. McClure, J. H. Smith, Multivariable cochain operations and little n-cubes, J. Am. Math. Soc. 16 (2003), pp. 681-704. 58

[33] J. E. McClure, J. H. Smith, Cosimplicial objects and little n-cubes, I, Amer. J. Math. 126 (2004), pp. 1109-1153. [34] J. E. McClure, J. H. Smith, Operads and cosimplicial objects: an introduction, J.P.C. Greenlees (ed), Axiomatic, Enriched and Motivic Homotopy Theory (2004), Kluwer Academic Pulishers, Netherlands, pp. 133-171. [35] R. J. Milgram, Iterated loop spaces, Ann. of Math. 84 (1966), pp. 286-403. [36] J. Milnor, Morse Theory, Annals of Mathematics Studies, No. 51, Princeton University Press, 1969. [37] J. Munkres, Topology, Pearson, 2nd edition, 2000. [38] J. Munkres, Elements of Algebraic Topology, West View Press, 1996. [39] P. Salvatore, Homotopy type of Euclidean configuration spaces, Suppl. Rendi- conti del Circolo Matematico di Palermo, Serie II, Suppl. 66 (2001), pp. 161-164. [40] D. Sinha, The (non-equivariant) homology of the little disks operad,available on the author’s webpage, 2010. [41] J. H. Smith, Simplicial goup models for ⌦nSnX, Isreal J. of Math. 66 (1989), 330-350. [42] D. E. Tamarkin, Another proof of M. Kontsevich formality theorem, Preprint math.QA/9803025. [43] D. E. Tamarkin, Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), pp. 65-72. [44] A. A. Voronov, Homotopy Gerstenhaber algebras, In Conférence MoshéFlato 1999, Vol. II (Dijon), volume 22 of Math. Phys. Stud., pp. 307-331. Kluwer Acad. Publ., Dordrecht, 2000. [45] B. Ward, Cyclic A structures and Deligne’s conjecture, Algebr. Geom. Topol. 12 (2012), no. 3, pp.1 1487-1551. [46] G. M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics 152, Berlin, New York, 1995. APPENDICES 59

A. Proof of Proposition 6.1.1

Proposition 6.1.1 follows from two lemmas.

n 2 n 1 2 Lemma A.0.2 Let S be the upper(lower)-hemisphere (x1, ,xn 1) R x1+ ± { ··· 2 2 n 1 n 2 + xn 1 =1,xn 1 0(xn 1 0) in R . Then S+ is a deformation retract of ···  } n 1 the unit ball D .

Proof Deﬁne n 1 n 1 H n 1 : D I D D ⇥ ! by

2 2 n 1 HD ((x1,x2, ,xn 1),t)=(x1,x2, ,xn 2, (1 t)xn 1 +t 1 x1 xn 2). ··· ··· ··· q n 1 It can be readily checked that HD is a well-deﬁned homotopy. Geometrically, 2 2 n 2 n 1 HD contracts each ﬁber over a point (x1, , 1 x1 xn 2)onS ··· ··· 2 2 n 2 n 1 to the point (x1, , 1 x1 xn 2)onS+ p. In addition, HD ( ,t)isthe ··· ··· · n 2 n 1 identity map on S pfor all t I.SoH n 1 is a deformation retraction from D + 2 D n 2 n 2 n 1 onto S+ (S+ is a deformation retract of D ).

3 2 Figure A.1. Deformation retraction from D to S+. 60

k n 1 Lemma A.0.3 There is a homeomorphism from P P P to D m1 ⇥ m2 ⇥···⇥ mk ⇥ k n 2 mapping Pm1 Pm2 Pmk ( i=2 v1 vi vk+1) homeomorphically onto S . ⇥ ⇥···⇥ ⇥ ··· ··· Proof Let be the geometricS center of P , l =1, ,k and F , j I the Cml b ml ··· jl l 2 l facets of P . Let = 1 (v + + v ). So is the barycenter of k. Then ml Ck k+1 1 ··· k+1 Ck k k+1 Pml = j I Fjl ml where is the join operation, and = i=1 v1 vi vk+1 k. l2 l ⇤C ⇤ ··· ··· C k So Pm1S Pmk = S ⇥ ···⇥ ⇥ b k k+1 (F ) (F ) (F ) (v v v ). j1 ⇤ Cm1 ⇥ ···⇥ jl ⇤ Cml ⇥ ···⇥ jk ⇤ Cmk ⇥ 1 ··· i ··· k+1Ck l=1 j I i=1 [ [l2 l [ Notice that each product on the right above has =( , , b, )asoneof C Cm1 ··· Cmk Ck its vertices and is the union of line segments from to a point of k (F ) C l=1 j1 ⇤ Cm1 ⇥

Fjl (Fjk mk ) (v1 vi vk+1 k) (Fj1 m1 ) S (Fjk mk ) ···⇥ ⇥ ···⇥ ⇤ C ⇥ ··· ··· C ⇤ C ⇥ ···⇥ ⇤ C ⇥ (v1 vi vk+1)andthelinesintersectonlyat .S ··· ··· b C b The line segments of di↵erent products having as a vertex above agree on the C intersections. So P P k is the union of line segments emanating m1 ⇥ ··· ⇥ mk ⇥ from and the union of the end points di↵erent from of the line segments is C C @(P P k). m1 ⇥ ···⇥ mk ⇥

1 2 Figure A.2. Decompositions of P2 and P2 P2 into line segments joined at their . ⇥ ⇥ ⇥ C

We will use a similar proof of that of Lemma 1.1 in [38] four times from now on. Here is the ﬁrst time. 61

k n 1 Let T : Pm1 Pmk R be the translation deﬁned by T (x)=x . ⇥ ···⇥ ⇥ ! C n 1 n 2 x Let r0 : R (0, , 0) S be the radial contraction given as r0(x)= x , \{ ··· } ! | | where x is the Euclidean norm of x. Since each half open ray emanating from | | k (0, , 0) intersects with T (@(P P )) at one and only one point, r0 ··· m1 ⇥ ···⇥ mk ⇥ k n 2 restricts to a continuous bijection r : T (@(P P )) S . Being the m1 ⇥ ···⇥ mk ⇥ ! continuous image of a compact space, T (@(P P k)) is compact, and m1 ⇥ ···⇥ mk ⇥ n 2 S is Hausdor↵, so r is indeed a homeomorphism.

k n 2 Now we extend r : T (@(P P )) S to R : T (P m1 ⇥ ···⇥ mk ⇥ ! m1 ⇥ ···⇥ k n 1 P ) D . Deﬁne mk ⇥ ! x 1 x if x =(0, , 0), r ( x ) 6 ··· R(x)=8 | | | | > (0, , 0) if x =(0, , 0). < ··· ··· Except for at x =(0, , 0), :>R is also continuous at x =(0, , 0). To see this, let 6 ··· ··· L be a lower bound of the Eulidean norm on @(P P k). Then for any m1 ⇥ ···⇥ mk ⇥ x L✏ ✏ > 0, if x

Since R is a continuous bijection from compact T (P P k) to Haus- m1 ⇥ ···⇥ mk ⇥ n 1 dor↵ D , R is indeed a homeomorphism. Furthermore, T is also a homeomorphism

k n 1 onto its image. So R T is a homeomorphism from P P to D m1 ⇥ ···⇥ mk ⇥ k n 2 mapping @(P P )homeomorphicallyontoS . m1 ⇥ ···⇥ mk ⇥

So far, P P ( k v v v )hasnotbeenmappedtothelower- m1 ⇥···⇥ mk ⇥ i=2 1 ··· i ··· k+1 n 2 hemisphere S . We will useS stereographic projection to achieve this. b

Recall k = v v v and @k =( k v v v ) (v v v v v v ). 1 2 ··· k+1 i=2 1 ··· i ··· k+1 1 2 ··· k 2 ··· k k+1 Then S S S b 1 1. k = k 1 (v2 + + vk)isinInt(v1v2 vk v2 vkvk+1). N ··· ··· ··· S 62

2. = 1 (v + v )isinInt( k v v v ). Sk 2 1 k+1 i=2 1 ··· i ··· k+1 S b 3. , and lie on the same line and : =2:(k +1). Nk Ck Sk |NkCk| |NkSk|

Figure A.3. i, i and , i =2, 3, 5. N C Si

Now we let =( , , , )and =( , , , ). So N Cm1 ··· Cmk Nk S Cm1 ··· Cmk Sk 1. is in the relative interior of @(P P k) Int(P P N m1 ⇥ ···⇥ mk ⇥ \ m1 ⇥ ···⇥ mk ⇥ ( k v v v )). i=2 1 ··· i ··· k+1 S b 2. is in the relative interior of P P ( k v v v ). S m1 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 S b 3. is in Int(P P k). C m1 ⇥ ···⇥ mk ⇥

4. , and lie on the same line and : =2:(k +1). N C S |N C| |N S| Let be an element in SO(n-1) rotating the vector so that it is aligned N C n 1 n 1 with the positive xn 1 axis. Notice that : D D is a homeomorphism. So ! b := R T is a homeomorphism.

n 1 n 1 Let N =(0, , 0, 1) R and S =(0, , 0, 1) R . The stereographic ··· 2 ··· 2 projections n 2 n 2 pN0 : S N R \ ! 63

x1 x2 xn 2 (x1, ,xn 1) ( , , , ) ··· 7! 1 xn 1 1 xn 1 ··· 1 xn 1 and n 2 n 2 pS0 : S S R \ ! x1 x2 xn 2 (x1, ,xn 1) ( , , , ) ··· 7! 1+xn 1 1+xn 1 ··· 1+xn 1 are homeomorphisms with inverses

2 2 1 2y1 2yn 2 1+y1 + + yn 2 p0N (y1, ,yn 2)=( 2 2 , , 2 2 , 2 ··· 2 ), ··· 1+y1 + + yn 2 ··· 1+y1 + + yn 2 1+y1 + + yn 2 ··· ··· ··· 2 2 1 2y1 2yn 2 1 y1 yn 2 p0S (y1, ,yn 2)=( 2 2 , , 2 2 , 2 ··· 2 ). ··· 1+y1 + + yn 2 ··· 1+y1 + + yn 2 1+y1 + + yn 2 ··· ··· ··· Since

k n 2 n 2 D := b(P P ( v v v )) S N S m1 ⇥ ···⇥ mk ⇥ 1 ··· i ··· k+1 ⇢ \ i=2 [ and b

k n 2 k n 2 D := b(@(P P ) Int(P P ( v v v ))) S S, N m1 ⇥···⇥ mk ⇥ \ m1 ⇥···⇥ mk ⇥ 1 ··· i ··· k+1 ⇢ \ i=2 [ n 2 b n 2 pN0 and pS0 restric to homeomorphisms pN and pS from DS and DN respectively to their images.

P P ( k v v v ) admits a presentation of the following m1 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 form S b k k (F ) (F ) (F ) (v v v v ). j1 ⇤Cm1 ⇥···⇥ jl ⇤Cml ⇥···⇥ jk ⇤Cmk ⇥ 1 ··· i j ··· k+1Sk l=1 j I i=2 j=1,k+1 [ [l2 l [ [ b b Each product on the right has =( , , , ) as one of its vertices and it S Cm1 ··· Cmk Sk is a union of line segments from to a point on k (F ) F S l=1 j1 ⇤ Cm1 ⇥ ···⇥ jl ⇥ ···⇥

(Fjk mk ) (v1 vivj vk+1 k) (Fj1 m1 ) S (Fjk mk ) (v1 vivj vk+1) ⇤C ⇥ ··· ··· C ⇤C ⇥···⇥ ⇤C ⇥ ··· ··· and the lines intersect only at . S b b S b b 64

The line segments of di↵erent products above agree on the intersections. So P P ( k v v v )istheunionoflinesegmentsemanating m1 ⇥ ··· ⇥ mk ⇥ i=2 1 ··· i ··· k+1 from and the unionS of the end points di↵erent from of the line segments is S b S k n 2 @(P P ( v v v )). Thus, p (D )istheclosureofan m1 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 N S n 2 n 2 open set in R containingS the origin and pN (DS )isaunionoflinesegmentsem- b n 2 anating from the origin such that each segment intersects @(pN (DS )) at only one point.

n 2 n 2 n 2 Therefore, we have a homeomorphism GS : pN (D ) D R obtained S ! ⇢ similar to that of R.

Every line segment x above and the point determine a half plane. This half S C plane intersects @(P P k)atapiecewiselinearpath y y x m1 ⇥ ···⇥ mk ⇥ N 1 ··· m S such that y y x is mapped to a line segment emanating from the origin in N 1 ··· m n 2 n 2 n 2 pS(DN ). Thus, pS(DN )istheclosureofanopensetinR containing the origin n 2 and pS(DN )isaunionoflinesegmentsemanatingfromtheoriginsuchthateach n 2 segment intersects @(pS(DN )) at only one point.

n 2 n 2 n 2 Therefore, we have a homeomorphism GN : pS(D ) D R obtained N ! ⇢ similar to that of R.

n 2 n 2 Now we deﬁne f : S S by ! 1 n 2 p GS pN (x),x D f(x)= N 2 S 8 1 n 2 p G p (x),x D . < S N S 2 N Notice that each branch of:f is continuous and they agree on the overlap. (See the ﬁgure below.) So f is continuous. Being a continuous bijection from compact

n 2 Hausdor↵ S onto itself, f is thus a homeomorphism.

n 2 n 2 n 1 n 1 Now we extend f : S S to F : D D by ! ! 65

Figure A.4. Two stereographic projections.

x f( ) x ,x=(0, , 0) x | | 6 ··· F (x)=8 | | > (0, , 0),x=(0, , 0). < ··· ··· Similar to R, F is continuous> at x =(0, , 0) because for any ✏ > 0, if x < ✏, : ··· | | x then F (x) F (0) = f( x ) x = x < ✏. Being a continuous bijection from compact | | | | | || | | | n 1 Hausdor↵ D onto itself, F is thus a homeomorphism.

k n 1 Therefore, F b is a homeomorphism from P P P to D m1 ⇥ m2 ⇥ ···⇥ mk ⇥ k n 2 mapping Pm1 Pm2 Pmk ( i=2 v1 vi vk+1) homeomorphically onto S . ⇥ ⇥···⇥ ⇥ ··· ··· k Thus, F b also maps @(Pm1 PmS2 Pmk ) Int(Pm1 Pm2 Pmk ⇥ ⇥ ···⇥ b ⇥ \ ⇥ ⇥ ···⇥ ⇥ k n 2 ( v v v )) homeomorphically onto S . i=2 1 ··· i ··· k+1 + S k b k Proof [of Proposition 6.1.1] Deﬁne HPm Pm :(Pm1 Pmk ) I 1 ⇥···⇥ k ⇥ ⇥ ···⇥ ⇥ ⇥ ! P P k by m1 ⇥ ···⇥ mk ⇥

1 1 k n 1 HPm Pm (x, t)=b F HD (F b(x),t). 1 ⇥···⇥ k ⇥ 66

B. Proof of Proposition 6.1.2

Proof [of Proposition 6.1.2] Let

n 2 n 1 n 2 S I✏ := (x1, ,xn 2,xn 1 ) R (x1, ,xn 1) S , 0 ✏ , ⇥ { ··· 2 ··· 2   } and the extended closed (n 1)-ball be

n 1 n 1 n 2 Ext✏(D ):=D (S I✏). ⇥ [ Deﬁne n 2 n 1 n 2 HS I✏ :(S I✏) I Ext✏(D ) ⇥ ⇥ ⇥ !

n 2 by HS I✏ ((x1, ,xn 1),t)= ⇥ ···

2 2 2 2 2 1 x1 xn 2 (x1, ,xn 2,xn 1 + t(xn 1 + 1 x1 xn 2 + ✏) ··· ). ··· ··· ✏ q p

n 2 Then HS I✏ is a well-defined homotopy. It linearly extends each fiber over ⇥ n 2 n 1 (x1, ,xn 2,xn 1 ✏)inS I✏ to the fiber in Ext✏(D ). For each t I, ··· ⇥ 2 n 2 HS I✏ ( ,t)isahomeomorphismontoitsimage. ⇥ ·

3 Figure B.1. The extended closed 3-ball Ext✏(D ). 67

By extending R and possibly perturbing F , we get R and F such that

k e e n 2 F R T :(Pm1 Pmk ( v1 vi vk+1)) I✏ S I✏ ⇥ ···⇥ ⇥ ··· ··· ⇥ ! ⇥ i=2 [ e e b is a homeomorphism whose images of P P ( k v v v ) 0 and m1 ⇥···⇥ mk ⇥ i=2 1 ··· i ··· k+1 ⇥{ } k n 2 n 2 Pm1 Pmk ( i=2 v1 vi vk+1) ✏ are S S and S (0, , 0, ✏) , ⇥ ···⇥ ⇥ ··· ··· ⇥ { } b { ··· } respectively. Furthermore,S F b (= F R T )andF R T agree on b P P ( k v v v ). m1 ⇥ ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 e e S b We deﬁne H k : (Pm Pm ( v1 v v )) I✏ 1 ⇥···⇥ k ⇥ i=2 ··· i··· k+1 ⇥ S k b ((P P ( v v v )) I ) I Ext (P P k) m1 ⇥ ···⇥ mk ⇥ 1 ··· i ··· k+1 ⇥ ✏ ⇥ ! ✏ m1 ⇥ ···⇥ mk ⇥ i=2 [ by b

1 1 1 1 H k (x, t)=T R F H n 2 (F R T (x),t). (Pm1 Pmk ( i=2 v1 vi vk+1)) I✏ S I✏ ⇥···⇥ ⇥ ··· ··· ⇥ ⇥ S b e e e e Notice that H k ( ,t)andH k ( ,t)agree Pm Pm (Pm Pm ( v1 v v )) I✏ 1 ⇥···⇥ k ⇥ · 1 ⇥···⇥ k ⇥ i=2 ··· i··· k+1 ⇥ · on P P ( k v v v )foreachtS I. Then we can deﬁne m1 ⇥ ··· ⇥ mk ⇥ i=2 1 ··· i ··· k+1 2 b k k k HExt✏(Pm Pm ) : ExtS✏(Pm1 Pmk ) I Ext✏(Pm1 Pmk ) 1 ⇥···⇥ k ⇥ ⇥ ···b ⇥ ⇥ ⇥ ! ⇥ ···⇥ ⇥ k k k by HExt✏(Pm Pm )(x, t)=HPm Pm (x, t)ifx Pm1 Pmk 1 ⇥···⇥ k ⇥ 1 ⇥···⇥ k ⇥ 2 ⇥ ···⇥ ⇥ and H k (x, t)=H k (x, t)ifx (P Ext✏(Pm Pm ) (Pm Pm ( v1 v v )) I✏ m1 1 ⇥···⇥ k ⇥ 1 ⇥···⇥ k ⇥ i=2 ··· i··· k+1 ⇥ 2 ⇥ P ( k v v v )) I . It canS be readily checked that the three ···⇥ mk ⇥ i=2 1 ··· i ··· k+1 ⇥ ✏ b conditions areS satisﬁed. b 68

C. How were the results discovered?

The results in this thesis were inspired by cellular homology computations. Below is part of a report documenting the discovery made on airplanes, trains and buses through half of China during the summer of 2013, which preceded the mathematical formulation in the main body of this work. So it should only be read as motivation, rough idea and inspiration.

C.1 A Motivating Example

The pair of spaces of the figure 8 and the letter ✓ is a popular example for il- lustrating the concept of homotopy equivalence. See Examples 2 and 3 on page 362 in [37], Example 3 from page 109 to page 110 in [38] and the right two figures in the first picture on page 2 in [18].

Figure C.1. The ﬁgure 8 and the ✓ spaces.

The method of showing 8 being homotopy equivalent to ✓ in the above references uses two deformation retractions from a larger space (the twice punctured plane) to 8and✓, respectively. By inspecting these two deformation retractions, we have the following maps f and g between ﬁgure 8 and ✓ directly. 69

Figure C.2. Direct homotopy equivalences between ﬁgure 8 and ✓.

The picture below shows that f and g are indeed homotopy inverses of each other.

Figure C.3. g f id ; f g id . ' 8 ' ✓

What if we do not know the above deformation retractions? Is there another way to ﬁnd f and g?Theanswerisyesandthismethodmovitatesourstudyofadirect equivalence from D2(n)toCact(n)forn =3inthenextsections.Letuslookatthe top-dimensional cellular homologies of these two spaces. Both H1(8) an H1(✓)are isomorphic to Z Z. But they are not equal to the aforementioned abstract abelian group. Rather, they have geometric content: H1(8) has two generators represented by two circles sharing a 0-cell while H1(✓)hastwogeneratorsrepresentedbytwo circles sharing a vertical edge. If f and g induce the isomorphisms (the left circle in 8ismappedtoandfromtheleftcirclein✓ and the right circle in 8 is mapped to and from the right circle in ✓), then it is natural to let f and g be deﬁned in the previous way. 70

Finding a homotopy equivalence between two spaces means we first need to have global understanding of these two spaces. But doing this at one step is in general very dicult. However, if the two spaces (CW complexes) have the same dimension, then we could first find simpler subspaces representing the generators (we will call these subspaces generators for simplicity later) of the top-dimensional cellular homologies of these two spaces, respectively. Then we compare these subspaces. If we are lucky, we could get useful information on finding homotopy equivalences.

Recall that D deformation retracts to an (n 1)-dimensional regular CW complex 2 (n)throughtheconﬁgurationspaceofn ordered distinct points in R2 and Cact F deformation retracts to Cact1(n), which is also (n 1)-dimensional. Furthermore, both (n)andCact1(n)containthesamenumberof0-cells(n! of them). So we are F in a particularly favorable situation.

C.2 Generators of H ( (3)) and H (Cact1(3)) 3 F 3

C.2.1 (n) and Cact1(n), n =1, 2, 3 F

Recall that the 3n-dimensional D2(n)deformationretractstothe2n-dimensional F (R2,n). In general, the conﬁguration space of n points in Rk is of dimension kn. These spaces admit di↵erent CW complex structures of lower dimensions. See [30] for the case of three points in Rk, and [39] for the case of n points in Rk for k 3.

Recall that by [7], F (R2,n) deformation retracts to (n), which is of dimension F n 1. We know (1) is a singleton. (2) is a circle consisting of two 0-cells and two F F 1-cells. 71

Figure C.4. (2). F

(3) consists of six 0-cells, twelve 1-cells and six 2-cells in the shape of hexagons F (permutohedra of order 3). So (3) is obtained by gluing 6 permutohedra of degree F 3alongtheirfaces.Its1-skeletonand2-cellswithchosenorientationsaresketched below.

Figure C.5. 1-skeleton of (3). F 72

Figure C.6. 2-cells of (3). Notice that all the 1-cells are drawn in order to show which are theF boundaries of each 2-cell.

On the other hand, Cact1(n)hasthestructureofaﬁniteCW complex also of dimension n 1. Cact1(1) is a single 0-cell. Cact1(2) is a circle consisting of two 0-cells and two 1-cells.

Figure C.7. Cact1(2).

Clearly, or from our description above, Cact1(n)and (n)arehomeomorphic F when n =1and2.

Cact1(3) is a more complicated complex. It has six 0-cells, eighteen 1-cells (twelve asymmetric ones and six rabbit heads) and twelve 2-cells (six triangles and six rectangles). Its 1-skeleton and 2-cells with chosen orientations are sketched in the next page. The total space is the union of them all. 73

Figure C.8. 1-skeleton of Cact1(3).

Figure C.9. The triangular 2-cells of Cact1(3). More 1-cells are shown in order to illustrate which 1-cells are the boundaries of each 2-cell.

Figure C.10. The rectangular 2-cells of Cact1(3). More 1-cells are shown in order to illustrate which 1-cells are the boundaries of each 2-cell. 74

C.2.2 Computation and comparison of the second homology generators.

Motivated by the homotopy equivalences between the ﬁgure 8 and ✓ spaces, we will compare the spaces Cact1(3) and (3) semi-globally by looking at their second F homology generators ﬁrst.

After solving a system of twelve linear equations with six unknowns, we ﬁnd that H ( (3)) is generated by two “tori” each with two singularities: two points on the 2 F inner cylinder are glued to two points on the outer cylinder. See Figure C.11. A third torus with two singularities is the ﬁrst minus the second.

1 By solving a system of eighteen linear equations with twelve variables, H2(Cact (3)) is found to be generated by the following two tori. See Figure C.12. Notice that the ﬁrst torus minus the second gives a third torus with the “hole” openning from the upper left to the lower right.

If we compare the generators of H (Cact1(3)) with those of H ( (3)), we ﬁnd 2 2 F that they are very alike, except the singularities. (Indeed, we have chosen the generators most similar in two spaces.) Notice that the outer cylinder of a generator of

1 H2(Cact (3)) is a union of two hexagons (each hexagon is the union of two triangles with a rectangle) and so is that for H ( (3)). The inner cyclinder of a generator of 2 F H (Cact1(3)) is a union of two rectangles while that of H ( (3)) is still a union of 2 2 F two hexagons. We actually would like to map a generator to a generator, but how to understand this discrepancy?

In fact, we can add a triangle and then subtract the same one to each edge (inter-

1 section of the two rectangles) of the inner cylinder of the torus for H2(Cact (3)). This doesn’t change the generators algebraically, but it does give a di↵erent topological 75

Figure C.11. The two generating singular tori of H ( (3)). 2 F

1 Figure C.12. The two generating tori of H2(Cact (3)).

picture: torus with two singular triangles. See Figure C.13.

Now we have a di↵erent presentation of Cact1(3): Six hexagons (permutohedra of order 3) each consisting of two triangles and one rectangle. If two permutohedra share a triangle, they are glued along this triangle. So Cact1(3) is obtained by gluing 6 permutohedra along the 6 triangles. See Figure C. 14. Notice that it is indistin- 76

1 Figure C.13. The same generators for H2(Cact (3)) with di↵erent topological pictures.

guishable from Figure C.6. [2], [10], [13] and [40] were the major references when the computations were carried out. For example, they conﬁrm that H ( (3)) and 2 F 1 H2(Cact (3)) are both isomorphic to Z Z.

Figure C.14. Permutohedral presentation of Cact1(3).

C.3 Homotopy equivalences f : Cact1(3) (3) : g. $ F

Now let us give the homotopy equivalences f : Cact1(3) (3) : g. $ F

g is “identity” on each hexagon. This map is 1 1ontheinteriorofthesquare ! region but 2 1ontheinteriorofthetrianglularregions. ! 77

f pushes each triangle in a hexagon to its outer two edges. At the same time, the remaining space is ﬁlled by the points stretched out from the rectangle. So for each rectangle, we get a hexagon. These hexagons only intersect on their faces. Then each hexagon is mapped “identically” to the corresponding hexagon in (3). F

We require that the six maps obtained by g restricted to the hexagons agree over the 1-skeleton. So do we require this condition for f.

Then f and g are homotopy inverses.

To see the proof, it is enough to look at the restrictions of f to a cylinder with two triangular wings1 (studied by Kaufmann in many of his articles on Cact and the Arc operad.) and g to a cylinder. It is readily seen that g f id and f g id. ' '

Figure C.15. Homotopy equivalences between Cact1(3) and (3). F

1I remember vividly that Professor Kaufmann showed the same space to me when I went to his MA572 Introduction to Algebraic Topology oce hour inquiring about abstract simplicial complexes in Spring 2011. 78

C.4 Do not start to do it unless you think it is obvious!

(4) has 4! = 24 0-cells, 3 4! = 72 1-cells, 3 4! = 72 2-cells (hexagons and F · · rectangles) and 4! = 24 3-cells.

Cact1(4) has 4! = 24 0-cells, 6 4! = 144 1-cells, 9 4! = 216 2-cells (rectangles and · · triangles) and 5 4! = 120 3-cells (cubes, prisms and tetrahedra). So it is not very · instructive to plot its skeletons.

By [7], a 3-cell in (4) is a permutohedron of order 4, or truncated octahedron. F The picture for one 3-cell is given below.

Figure C.16. Permutohedron of order 4. It can be realized as the regular octahedron with six pyramids cut o↵.

No attempts of computing H3( (4)) or H3(Cact1(4)) were made. The n =3case F already gave us hints to look for the subspaces of Cact1(n)consistingofcertaincacti configurations whose tree order is compatible with something. Surprisingly, as we 79 saw from the picture below, fifteen 3-cells of Cact1(4) (one cube, eight prisms and six tetrahedra) form a permutohedron of order 4 (the squares of the permutohedron are rendered in pink or purple). This prompts us to redefine Cact1(4) as the space obtained by gluing the 24 permutohedra along the constituent tetrahedra and prisms. On the other hand, (4) is obtained by gluing 24 permutahedra along their faces. F

Figure C.17. Permutohedron of order 4 inside Cact1(4).

Remark. The story is as follows. When I drew the 2-skeleton (consisting of closed 2-cells whose trees have partial orders compatible with a chosen total order ) cell by cell, the polyhedral surface closes back onto itself. I get an S2! I have never seen the permutohedron P4 before. After I googled the picture of it, I couldn’t hold my breath! The polytope my closed surface bounds is P4! 80

Now let us give the homotopy equivalences f : Cact1(4) (4) : g. $ F

g : (4) Cact1(4) sends each permutohedron in (4) to the associated permu- F ! F tohedron in Cact1(4). This map is 1 1ontheinteriorofthecube,2 1onthe ! ! interior of each of the two prisms with no color face rendering, 3 1ontheinterior ! of each of the six prisms with color face and 6 1ontheinteriorofeachofthe ! tetrahedra.

Before we deﬁne f : Cact1(4) (4), let us point out that there is only one way ! F for each prism or tetrahedron to be glued to other 3-cells to form a permutohedron of order 4. This makes the following f well-deﬁned.

First, let us deﬁne a homotopy H : Cact1(4) I (4). H is constructed in ⇥ ! F three stages.

When t [0, 1 ], the prisms without color face renderings are pushed to their 2 2 3 dimensional roof2 boundaries with the left space ﬁlled by points in the cubes.

When t [ 1 , 2 ], the six prisms with color face renderings and the six tetrahedra 2 3 3 are pushed to their 2 dimensional roof boundaries with the left space ﬁlled by points of the previous space.

When t [ 2 , 1], the triangles are pushed to their roof boundaries and the left 2 3 space are ﬁlled with points from the rectangles.

Examples of the two stage homotopy for the terahedra are illustred below.

2The dual graph (tree) of a roof has no black vertices (other than the root) of arity more than 1. 81

Finally, let f = H( , 1). ·

By the construction, g f id and f g id. ' '

Remark.

It was found out later that the step when t [ 1 , 2 ] should be split into 2 steps. • 2 3 3 This is because the prisms have initial branching number 2 while the tetrahedra have initial branching number 3. And the homotopy for the prisms should be done before the homotopy for the tetrahedra.

It was not clear back then how to do the homotopy. In fact, looking at the • tetrahedra and prisms themselves is not a correct viewpoint. (Looking at products of simplices is even worse.) One should look at products of permutohedra P , ,P and a simplex k which parametrizes the cacti conﬁgurations with m1 ··· mk k sub cacti conﬁgurations with m , ,m lobes, respectively, over the root lobe. 1 ··· k The technique of extended products, which is a local tool, solved this problem.

It was not clear back then how to make the ﬁrst and last step well-deﬁned. Later • it was found that the iterated cones construction solved the problem.

Cact1(n)and (n), especially when n 5, are hard to visulize. But after the F previous exploration, the following vague theorem was formulated. Its “proof” should not be counted as a proof. But it contains the essential ideas. 82

Theorem C.4.1 For n 1, (n) is a space obtained by gluing permutohedra of order F n along their proper faces and Cact1(n) is a space obtained by gluing permutohedra of order n along some of the constituent polyhedra of dimension n 1. They are homotopy equivalent and the homotopy inverses can be given explicitly.

Proof That each (n 1) cell of (n)isapermutohedronofordern follows from F the combinatorial nature of (n). For Cact1(n), consider a cell having the topolog- F ical type of a tree whose vertices have arity one except the leaf vertex. This cell is an (n 1)-dimenional cube. Now we will resort to a physics proof (So be cautious!).

Pick the center element from the above cell. Treat it as a physical conﬁguration of n elastic coins each touching the one below it with the bottom one pinned to the background. Let i ,i , ,i be the labels of the coins from the top to the bottom. 1 2 ··· n Imagine this coin conﬁguration lies on a vertical plane and they are under the grav- itational forces. Now we give the system a small disturbance. Then this topological type changes into other topological types. (In the language we formulated in the main text, this says we are looking at all the trees in i1i2 in )Weglueallthecells(onefor T ··· each possible generated topological type) along their faces. Checking the geometric picture (Well, only the pictures for n =1, 2, 3, 4wereavailable.Sothisstepisonly a wish.), we obtain a space “identical” (homeomorphic) to an (n 1) cell of (n) F whose disjoint points are labelled by i ,i , ,i from top to bottom. 1 2 ··· n

Then Cact1(n)isaspaceofthesepermutohedraofordern glued along the other constituent polyhedra of dimension n 1thanthecubeinthecenter.

Now let us give the homotopy inverses f : Cact1(n) (n):g. Restriction of $ F g : (n) Cact1(n)oneachofthen! permutohedra is an “indentity”. This map is F ! multi-to-one except in the interior of each of the n! cubes of dimension n 1. 83

f : Cact1(n) (n)istheendingresult(t =1)ofahomotopyH done in stages. ! F

H ﬁrst pushes the each of the cubes of dimension n 1tooccupythespaceof the permutohedron of order n while the other constituent polytopes of dimension n 1arepushedtotheirroofs. Remark. This last part was a mistake. As we commented before, we should not look at these polytopes individually. We need to look at colim [l]C in the language formulated in the main text. T

Then H does the pushing to the n 2cellsonthesurfaceofeachpermutohedron of order n to form permutohedra of order n 1.

Then H does the pushing to the n 3cellsonthesurfaceofeachpermutohedron of order n to form permutohedra of order n 2.

The process continues until permutohedra of order 3 are generated.

Because each polytope has only one way to be pushed to its roof, H and hence f are well deﬁned.

Physically, what we have at the beginning are flat plastic bags which contins air only in their centers. What H does is to inflate them in the highest dimension, and then to inflate the boundaries, and then to inflate the boundaries of the boundaries ... until dimension 2 boundaries (hexagons) are inflated.

From our construction, f and g are homotopy inverses. VITA 84

VITA

Yongheng Zhang was born in Baoji, Shaanxi Province, China, three days before the opening ceremony of the Los Angeles Summer Olympics. He loved mathematics when he was little, but he gradually lost his interest in this subject since he went to high school. After the 2003 national higher education entrance examination, he chose automation as his undergraduate major in order to avoid mathematics. He lived in Changsha, Hunan Province, China for three years attending Central South University and then did his senior design project at Purdue University Calumet, Hammond, Indiana, where he continued his study of engineering as a graduate student. In his six years of study of engineering, he realized two things: (1) mathematics is fundamental to physical (and even social) sciences and (2) mathematics itself is worth studying even with no agenda for application. So after he got a master degree, he went to Purdue University, West Lafayette, Indiana to study mathematics. He worked on algebraic topology under the direction of Ralph M. Kaufmann and will get his Ph.D. degree in May, 2015. He will continue his research and teaching as a visiting assistant professor in mathematics at Amherst College, Amherst, Massachusetts.