aa r X i v : . [ m a t h . A T ] A ug Cubical Acyclic Homotopy Excision
Kay Werndli, EPF Lausanne, 24 th September, 2018
Abstract
Given a strong homotopy pushout cube of spaces A , we measure how far it is fromalso being a homotopy pullback cube. Explicitly, letting P be the homotopy colimit of thediagram obtained from A by forgetting the initial vertex A ∅ , we study the homotopy fibreof the double suspension of A ∅ → P . This difference is expressible in terms of the homotopyfibres of the original maps in A .
0. Introduction
There is a classical duality between homotopy and homology in that homotopy groups arecompatible with homotopy pullbacks, while homology groups are compatible with homotopypushouts. In both cases, we obtain long exact sequences and the classical homotopy excisiontheorem (which has been proven and generalised time and again by different people [1; 3; 10; 12])answers the question within which range a homotopy pushout induces a long exact sequence ofhomotopy groups rather than homology groups.In [6] and [9], this classical mere connectivity result for squares is extended to a cellularand acyclic inequality, respectively, which recovers the connectivity statement but potentiallyallows one to draw broader consequences than just being able to identify an initial range ofvanishing homotopy groups. For example, the stronger acyclic inequality is essential in [7],where it is used to prove a generalisation of “Bousfield’s key lemma” [2] in order to describecellular properties of Postnikov sections and spaces in Farjoun’s modified Bousfield-Kan tower.The classical homotopy excision also has analogues for higher-dimensional cubical di-agrams ([3; 10; 12] or the more recent treatment in [13]) and in this short article, we take afirst step towards strengthening these, again obtaining stronger acyclic inequalities, rather thanmere connectivity statements. Explicitly, our main result is the following.(0.1)
Theorem.
Let A : ◻ → sSets be a strong homotopy pushout cube of connectedspaces, with homotopy fibres F k := hFib( A ∅ → A k ) and comparison map q : A ∅ → holim ⌟ A .As long as the homotopy fibres F , F and F are again connected,hFib(Σ q ) > Σ(Ω F ∗ Ω F ∗ Ω F ) . Restricting our attention to just connectivities and using the Hurewicz theorem, werecover the classical cubical homotopy excision theorem [12] for simply connected spaces.(0.2)
Corollary. If A : ◻ → sSets is a strong homotopy pushout cube of simply connectedspaces whose homotopy fibres hFib( A ∅ → A k ) are i k -connected for i k >
1, then the total fibrehFib( q : A ∅ → holim ⌟ A ) is ( i + i + i )-connected.
1. Setup and Notation
We will mainly follow the notations and conventions established in [9]. Our base category in thisarticle is the category sSets of (unpointed) simplicial sets, equipped with the standard Quillen(or Kan) model structure. Hence “space” will mean “simplicial set” throughout. As for cubicaldiagrams, given n ∈ N , we write h n i := { , . . . , n } , ◻ n := P h n i , ⌜ n := ◻ n \ {h n i} , ⌟ n := ◻ n \ { ∅ } ection 2. Sets of Spaces n -cube, and the standard indexing posets for n -dimensional pushouts and pull-backs respectively. Now a cubical diagram (of spaces) is just some diagram X : P → sSets indexed by a poset P ∼ = ◻ n . Given such a P ∼ = ◻ n as well as p q in P , we define ∂ qp := { r ∈ P | p r q } and if X : P → sSets is a cubical diagram, we denote its facesby ∂ qp X := X | ∂ qp . As particularly important special cases, given a single element p ∈ P , anddenoting the bottom and top elements of P by ⊥ (e.g. ⊥ = ∅ for P = ◻ n ) and ⊤ (e.g. ⊤ = h n i for P = ◻ n ), respectively, we write ∂ p := ∂ ⊤ p and ∂ p := ∂ p ⊥ . Similarly for ∂ p X and ∂ p X . Forthe standard n -cube ◻ n we allow ourselves the notational abuse of omitting braces in the faceindices. So, for example, we will write ∂ , instead of ∂ { , } .Since every P ∼ = ◻ n is a complete lattice, we use standard lattice notation, such as ⊥ forthe bottom element, ⊤ for the top element and ¬ for the complement. Again, for the standard n -cube ◻ n , we abusively write ¬ k or ˆ k for h n i \ { k } instead of ¬{ k } .As is well-known, higher-dimensional homotopy pushouts (and dually pullbacks) canbe calculated inductively as a series of ordinary two-dimensional homotopy pushouts. Moreexplicitly, given a diagram X : ⌜ n +1 → sSets thenhocolim ⌜ n +1 X ≃ hocolim (cid:16) hocolim ⌜ n ∂ n +1 X ← hocolim ⌜ n ∂ ¬ n +1 X → X ¬ n +1 (cid:17) , which follows immediately from Thomason’s theorem [8, Theorem 26.8] and the representationof ⌜ n +1 as a Grothendieck construction ⌜ n +1 ∼ = Z ⌜ (cid:16) ⌜ n ← ⌜ n → {∗} (cid:17) . Of course, since everything is symmetrical, it is not essential which pair of opposing faces isused for this calculation and we only chose ∂ n +1 and ∂ ¬ n +1 as an example. In fact, it will playa crucial role later that we can pick any pair of opposing faces.We can now reapply this to the two ⌜ n in the above Grothendieck construction etc. Inthe end, we can represent ⌜ n +1 by a Grothendieck construction involving only ⌜ and points.One can use this to show that if all 2-faces of a cubical diagram are homotopy pushouts, thenall higher-dimensional faces (as well as the entire cube) are homotopy pushouts. Such diagramsare called strong homotopy pushouts [12]. Dually for strong homotopy pullbacks .
2. Sets of Spaces
Our basic objects of study are sets of spaces. Given two such sets of spaces M , N , we write N > M (“ N is killed by M ”), which we call an acyclic inequality and means that every spacein N becomes contractible upon left Bousfield localisation of sSets at { A → ∆[0] | A ∈ M } .If M or N are (equivalent to) singletons, we will usually omit the surrounding braces.(2.1) Example.
As one can easily check, letting S − be the empty space, if S − ∈ M then N > M for all N . Just as easily, one checks that N > S iff all X ∈ N are non-empty. Moregenerally, one can show [11] that N > S n +1 iff every X ∈ N is n -connected.Equivalently [4; 5], N > M can be taken to mean that M is contained in the smallestclass ¯ C ( N ) containing N and closed under weak equivalences, (unpointed) homotopy colimitsindexed by contractible categories and extensions by fibrations (whenever F → E → B is a fibresequence for any base point of B and B , F ∈ ¯ C ( N ) then also E ∈ ¯ C ( N )).More restrictively, if we let C ( N ) be the smallest class containing N and closed underjust weak equivalences and homotopy colimits indexed by contractible categories, we define N ≫ M to mean that M ⊆ C ( N ) [4; 5; 11]. Obviously, N ≫ M implies N > M . ection 2. Sets of Spaces f : E → B , and b ∈ B , let us write hFib b ( f ) for the homotopy fibre of f above b (or hFib ∗ ( f ) if b is clear from the context). On the other hand, we also writeh F ib( f ) := { hFib b ( f ) | b ∈ B } . As a final notation for the homotopy fibre, we simply write hFib( f ) to indicate that all elementsof h F ib( f ) are weakly equivalent (e.g. if B is connected) and that we just pick one representative.(2.2) Remark.
With this notation, the closure of a cellular class C ( A ) under extensions byfibrations can be stated without mentioning base points. If C is a closed class (i.e. closed underweak equivalences and pointed homotopy colimits), it is closed under extensions by fibrationsiff for every map f : E → B , having B ∈ C and h F ib( f ) ⊆ C implies E ∈ C .Since we are working with unpointed spaces, taking loop spaces is not well-defined andour convention is that Ω B := S − for B not connected. Otherwise, the homotopy type of theloop space is independent of the base point and we pick a representative. This means that Ω B is only defined up to homotopy and not functorial. The usual loop space with respect to some b ∈ B will by denoted by Ω b B or Ω ∗ B if the base point is clear from the context.Finally, two sets of spaces M , N are weakly equivalent iff every X ∈ M is weaklyequivalent to some Y ∈ N and vice versa. Also, when applying homotopical constructions tosets of spaces, it is always understood that these are to be applied elementwise. For instance, if M , N are sets of spaces, their join is M ∗ N := { A ∗ B | A ∈ M, B ∈ N } .Lots of results, which are classically only formalised for single pointed or connectedspaces generalise directly to arbitrary spaces when formalised using sets of spaces. Let us namea few of the most important ones here, which we are going to use. For more details see [15].Every composable pair of maps f : A → B , g : B → C and every base point b ∈ B givesrise to a fibre sequence hFib b ( f ) → hFib g ( b ) ( g ◦ f ) → hFib g ( b ) ( g ). When doing away with basepoints and taking sets of spaces, it doesn’t make sense anymore to speak of fibration sequencesbut we still have an acyclic inequality(2.3) h F ib( g ◦ f ) > h F ib( f ) ∪ h F ib( g ) . As a special case of a composable pair, if a map s : A → B has a retraction r , one obtains thathFib b ( s ) ≃ Ω ∗ hFib r ( b ) ( r ) for every b ∈ B . When taking fibre sets instead, we can no longerhope for a weak equivalence (just consider s : ∆[0] → S ) but only cellular inequalities:(2.4) h F ib( r ) ≫ h F ib( s ) ≫ Ω h F ib( r ) . We will need the following result [11, Appendix HL] about the commutation of homo-topy colimits and homotopy fibres on several occasions.(2.5)
Theorem. (Puppe)
Given a diagram X : I → sSets and τ : X ⇒ K a transformationto a constant diagram thenhFib k (cid:0) hocolim τ : hocolim X → K (cid:1) ≃ hocolim(hFib k τ )for every base point k of K .(2.6) Corollary. If X : ◻ n → sSets is a strong homotopy pullback, the homotopy fibre ofthe comparison map q : hocolim X | ⌜ n → X h n i above x ∈ X h n i ishFib x ( q ) ≃ n ˚ i =1 hFib x ( X ˆ ı → X h n i ) , whence h F ib( q ) ≃ n ˚ i =1 h F ib (cid:0) X ˆ ı → X h n i (cid:1) . ection 3. Thomason Magic Theorem. (Chachólski)
Every homotopy pushout square A f / / g (cid:15) (cid:15) B (cid:15) (cid:15) C g / / D gives rise to a cellular inequality h F ib( h ) ≫ h F ib( f ).Finally, we also need the unsuspended square case of the acyclic homotopy excisiontheorem, as established in [9].(2.8) Theorem.
Given a homotopy pushout square as in Chachólski’s theorem above withcomparison map q : A → holim( B → D ← C ), thenh F ib( q ) > Ω h F ib( f ) ∗ Ω h F ib( g ) .
3. Thomason Magic
In this section, we generalise a trick due to Chachólski [6], who shows that, given a transformationof spans
B A o o / / f (cid:15) (cid:15) CB A ′ o o / / C with hocolim P g (cid:15) (cid:15) Q, one has h F ib(Σ f ) ≫ h F ib( g ) ≫ Σ h F ib( f ). The way he goes about the first cellular inequalityis as follows:(a) The suspension map Σ f can be obtained as the homotopy colimit of∆[0] B o o / / B A o o / / f (cid:15) (cid:15) C C o o / / ∆[0]∆[0] B o o / / B A ′ o o / / C C o o / / ∆[0] , (we can take the homotopy pushouts of the four outer appendages first, by Thomason’stheorem [8, Theorem 26.8]). Instead, we can take the central homotopy pushouts first,and h F ib(Σ f ) ≫ h F ib( g ) then follows from Dror Farjoun’s theorem [4, Theorem 9.1]because all remaining fibres are contractible.(b) Similarly, g can be obtained as the homotopy colimit of B A ′ o o / / A ′ A o o / / f (cid:15) (cid:15) A ′ A ′ o o / / CB A ′ o o / / A ′ A ′ o o / / A ′ A ′ o o / / C. Again, we can instead take the central pushout first. By Puppe’s theorem, one hashFib ∗ (cid:0) hocolim( A ′ ← A → A ′ ) → A ′ (cid:1) ≃ hocolim (cid:0) ∆[0] ← hFib ∗ ( f ) → ∆[0] (cid:1) for every base point of A ′ and so hocolim( A ′ ← A → A ′ ) → A ′ is a fibrewise suspension of f . All in all, the induced map between the central pushouts has Σ h F ib( f ) as itsfibre set and h F ib( g ) ≫ Σ h F ib( f ) again follows from Dror Farjoun’s theorem. ection 4. Serre’s Theorem Thomason magic . The adding ofappendages is done by the following Grothendieck construction.(3.1)
Definition.
Given a poset P with a bottom element ⊥ , we defineApp( P ) := Z P F where F : P → Cat , p ( { ∅ } p = ⊥ ⌜ p = ⊥ with F ( ⊥ p ) : { ∅ } → ⌜ being the element { } for p = ⊥ and all other morphisms beingidentities. This comes with an inclusion P ֒ → App( P ), ⊥ 7→ ( ⊥ , ∅ ) and p ( p,
1) for p = ⊥ .With this construction at hand, one can generalise the above proof in a straightforwardmanner even to arbitrary posets with a bottom element (see [15] for more details).(3.2) Proposition. (Thomason Magic)
Given a transformation τ : X ⇒ Y of diagrams X , Y : ⌜ n → sSets such that every component τ S with S = ∅ is a weak equivalence, then(a) h F ib (cid:0) hocolim τ : hocolim X → hocolim Y (cid:1) ≫ Σ n − h F ib( τ ∅ : X ∅ → Y ∅ );(b) h F ib(Σ n − τ ∅ : Σ n − X ∅ → Σ n − Y ∅ ) ≫ h F ib (cid:0) hocolim τ : hocolim X → hocolim Y (cid:1) .
4. Serre’s Theorem
Another result we need to generalise to higher dimensions is the result [6, Theorem 7.1], whichis referred to as the generalised Serre theorem in op. cit .(4.1) Lemma.
Let τ : X ⇒ Y be a transformation of diagrams X , Y : ⌜ n → sSets and m ∈ N n such that τ | ◻ m − : X | ◻ m − ⇒ Y | ◻ m − viewed as a diagram ◻ m → sSets is a homotopy pushoutand such that τ is a weak equivalence outside of ◻ m − . Then τ ∗ : hocolim ⌜ n X → hocolim ⌜ n Y is a weak equivalence. Proof. If n <
2, the claim is trivial. The case n = 2 is in the proof of [6, Theorem 7.1] andfollows from a Fubini-type argument for homotopy colimits. The general case is by inductionon n . We use Thomason’s theorem [8, Theorem 26.8] and gethocolim X (cid:15) (cid:15) ≃ hocolim (cid:16) hocolim ⌜ n − X | ∂ n ≃ (cid:15) (cid:15) hocolim ⌜ n − X | ∂ ¬ n o o / / (cid:15) (cid:15) X ¬ n (cid:17) (cid:15) (cid:15) hocolim Y ≃ hocolim (cid:16) hocolim ⌜ n − Y | ∂ n hocolim ⌜ n − Y | ∂ ¬ n o o / / Y ¬ n (cid:17) , where the solid arrow on the left is a weak equivalences because ∂ n ⊆ ◻ n \ ◻ m − . If m < n , thesame applies to the right arrow, while the middle one is a weak equivalence by the inductivehypothesis. Finally, if m = n , then, by the claim’s hypotheses and Thomason’s theorem, theright-hand square in the diagram is a pushout and the claim follows from the case n = 2. (cid:3) ection 4. Serre’s Theorem Theorem. (Generalised Serre Theorem)
Let τ : X ⇒ Z be a natural transforma-tion of diagrams X , Z : ⌜ n → sSets and m ∈ N n such that τ | ◻ m − : X | ◻ m − ⇒ Z | ◻ m − viewed as a diagram ◻ m → sSets is a strong homotopy pullbackand such that τ is a weak equivalence outside of ◻ m − . Furthermore, let us fix any base pointin X ∅ and denote the fibres of the strong homotopy pullback τ | ◻ m − by F , . . . , F m ; i.e. F k := hFib ∗ ( X ∅ → X k ) for k < m and F m := hFib ∗ ( X ∅ → Z ∅ ) . Then, hFib ∗ (hocolim X → hocolim Z ) ≫ Σ n − m ( F ∗ . . . ∗ F m ). Proof.
The case n = 2 is [6, Theorem 7.1]. For a general n , we factor τ as X ⇒ Y ⇒ Z , where Y agrees with Z everywhere except at Y h m − i , which is such that τ | ◻ m − : X | ◻ m − ⇒ Y | ◻ m − viewed as a diagram ◻ m → sSets is a homotopy pushout.By the lemma above, the induced morphism hocolim X → hocolim Y is a weak equivalence andso it suffices to show thathFib ∗ (hocolim Y → hocolim Z ) ≫ Σ n − m ( F ∗ . . . ∗ F m ) .Y h m − i → Z h m − i is the comparison map for the strong homotopy pullback X | ◻ m − ⇒ Z | ◻ m − ,Puppe’s theorem allows us to identify its fibre as being F ∗ . . . ∗ F m . So, what we are reallygoing to show is thathFib ∗ (hocolim Y → hocolim Z ) ≫ Σ n − m hFib ∗ (cid:16) Y h m − i → Z h m − i (cid:17) . We now repeatedly use Thomason’s theorem to move this comparison map into a central locationof some subcube and then apply Thomason magic. Identifying ⌜ n ∼ = R ⌜ ( ⌜ n − ← ⌜ n − → {∗} ),we use Thomason’s theorem to write the (induced map between) homotopy colimits ashocolim Y (cid:15) (cid:15) ≃ hocolim (cid:16) hocolim ⌜ n − Y | ∂ (cid:15) (cid:15) hocolim ⌜ n − Y | ∂ ¬ o o / / ≃ (cid:15) (cid:15) Y ¬ (cid:17) ≃ (cid:15) (cid:15) hocolim Z ≃ hocolim (cid:16) hocolim ⌜ n − Z | ∂ hocolim ⌜ n − Z | ∂ ¬ o o / / Z ¬ (cid:17) . Observing that Y and Z agree everywhere except at h m − i , the two solid arrows on the rightare weak equivalences and using Dror-Farjoun’s theorem [4, Theorem 9.1], it suffices to showhFib ∗ (cid:16) hocolim ⌜ n − Y | ∂ → hocolim ⌜ n − Z | ∂ (cid:17) ≫ Σ n − m hFib ∗ (cid:16) Y h m − i → Z h m − i (cid:17) . For this, we simply repeat the above, identify ⌜ n − ≃ R ⌜ ( ⌜ n − ← ⌜ n − → {∗} ) and use Thoma-son’s theorem again etc. We do this m − (cid:0) Y | ∂ (cid:1) | ∂ = Y | ∂ , = Y | ∂ h i (and similarly for ¬ Z and higher order restrictions), we end up needing to show thathFib ∗ (cid:16) hocolim ⌜ n − m +1 Y | ∂ h m − i → hocolim ⌜ n − m +1 Z | ∂ h m − i (cid:17) ≫ Σ n − m hFib ∗ (cid:16) Y h m − i → Z h m − i (cid:17) . But now h m − i is the initial vertex of ∂ h m − i and the claimed cellular inequality is exactlywhat we get from the Thomason magic (3.2). (cid:3) ection 5. Suspended Comparison Map
5. Suspended Comparison Map
Starting with a strong homotopy pushout A : ◻ n → sSets , consider the comparison map q : A ∅ → P ∅ , where P : ◻ n → sSets agrees with A everywhere except at ∅ , where it is thehomotopy pullback of A | ⌟ n (possibly replacing A fibrantly first). From Thomason magic (3.2),we know thath F ib(Σ n − q ) ≫ h F ib (cid:18) A h n i s −→ hocolim ⌜ n P (cid:19) and so it suffices to understand this fibre set. Restricting the obvious natural transformations A ⇒ P ⇒ Const A h n i to ⌜ n and taking homotopy colimits, we get that s : A h n i → hocolim ⌜ n P has a retraction r : hocolim ⌜ n P → A h n i . But by (2.4) h F ib( s ) ≫ Ω h F ib( r ). Now, let S : ◻ n → sSets be the strong homotopy pullbackassociated to A (i.e. S := R Kan( A | h n i ⊲ ), where h n i ⊲ ⊂ ◻ n is the subposet of all ˆ k → h n i .Composing the two obvious transformations P ⇒ S and S ⇒ Const A h n i , restricting to ⌜ n andtaking homotopy colimits, we get a factorisationhocolim ⌜ n P p −→ hocolim ⌜ n S p ′ −→ A h n i of r : hocolim ⌜ n P → A h n i . The acyclic inequality (2.3) associated to this composable pair is(5.1) h F ib( p ′ ◦ p ) > h F ib( p ) ∪ h F ib( p ′ ) . Finally, Puppe’s theorem (2.5) together with Chachólski’s theorem (2.7) yieldh F ib( p ′ ) ≃ n ˚ k =1 h F ib (cid:0) A ˆ k → A h n i (cid:1) ≫ n ˚ k =1 F ( A ∅ → A k ) . (5.2) Proposition.
Let A : ◻ n → sSets be a strong homotopy pushout with correspond-ing homotopy pullback P , strong homotopy pullback S and comparison maps q : A ∅ → P ∅ , p : hocolim ⌜ n P → hocolim ⌜ n S . Then we have a cellular inequalityh F ib(Σ n − q ) > Ω (cid:18) h F ib( p ) ∪ n ˚ k =1 h F ib( A ∅ → A k ) (cid:19) . (cid:3)
6. The Main Theorem
To better understand the map p from proposition (5.2), we do not form the strong homotopypullback S directly. Instead, we construct a sequence of homotopy pullback cubes(6.1) P ≃ P (0) ⇒ P (1) ⇒ P (2) ⇒ P (3) ≃ S, where each transformation induces a map between homotopy colimits over ⌜ that we canunderstand. These three cubes correspond to the three pairs of opposite faces in ◻ . In thisway, we subsequently replace all possible 2-faces by homotopy pullbacks and therefore eventuallyarrive at a strong homotopy pullback.(6.2) Definition.
Given a cubical diagram X : ◻ n → sSets such that all X M with M = ∅ are connected and k ∈ h n i , we define F Xk := n hFib (cid:16) X M → X M ∪{ k } (cid:17) (cid:12)(cid:12)(cid:12) M ⊆ h n i \ { k } o (where we omit the top index “ X ” if it is clear from the context). We call this the collective k th (homotopy) fibre of X . ection 6. The Main Theorem Example. If X is a strong pushout, from Chachólski’s theorem (2.7), we know that F k ≫ F k (and clearly F k ≫ F k ). So, from a cellular viewpoint, F k collapses to a single fibre.(6.4) Lemma.
Given a fibre sequence F → E → B where F > A and
E > Σ A for somesimplicial set A , then B > Σ A . Proof.
We form the cofibre sequence E → B → B (cid:12) E . Now B (cid:12) E ≫ Σ F by [5, Proposition10.5] and h F ib( B → B (cid:12) E ) ≫ E by (2.7), from which the claim follows. (cid:3) (6.5) Proposition.
Let A : ◻ → sSets be a strong homotopy pushout of connected spacesand F k := hFib( A ∅ → A k ). If P : ◻ n → sSets is the corresponding homotopy pullback, then F Pk > ΣΩ F k for all k ∈ h i as long as F , F and F are connected. Proof.
Without loss of generality, let’s assume k = 1. The collective fibres F A and F P arealmost the same, with the former being killed by F . The only fibre in F P that is not in F A is hFib( P ∅ → A ). Picking any base-point on A ∅ makes everything (and in particular P ∅ )pointed. We now consider the fibre sequencehFib( A ∅ → P ∅ ) −→ hFib( A ∅ → A ) | {z } F −→ hFib( P ∅ → A )associated to A ∅ → P ∅ → A . Because F ≫ ΣΩ F (see [5, Corollary 10.6]) and using thelemma above, it suffices to show that hFib( A ∅ → P ∅ ) > Ω F . For this, we recall that P ∅ fitsinto a homotopy pullback square (of solid arrows) A ∅ / / P ∅ f / / (cid:15) (cid:15) holim( A → A , ← A ) p (cid:15) (cid:15) A g / / holim( A , → A , , ← A , ) . The homotopy fibre of p above any base-point of holim( A , → A , , ← A , ) is just thehomotopy pullback G := holim (cid:0) hFib( A → A , ) → hFib( A , → A , , ) ← hFib( A → A , ) (cid:1) . Since A is a strong homotopy pushout, all these fibres are killed by F , which is connected, andhence connected themselves. The fibre sequence associated to the homotopy pullback G isΩ hFib( A , → A , , ) → G → hFib( A → A , ) × hFib( A → A , )and since both the base and the fibre are killed by Ω F , it follows that also G > Ω F > S , sothat π ( G ) = ∅ . Since the base-point was arbitrary, it follows that p hits all components of thebase and hence h F ib( g ) ≃ h F ib( f ). From acyclic homotopy excision for squares (2.8), we knowthat h F ib( g ) > Ω hFib( A → A , ) ∗ Ω hFib( A → A , ) > Ω F ∗ Ω F > ΣΩ F , where we used that F is connected. By the same argument, we also have thath F ib (cid:0) A ∅ → holim( A → A , ← A ) (cid:1) > Ω F ∗ Ω F > ΣΩ F . Now, the acyclic inequality (2.3) associated to the top composite in the last diagram giveshFib( A ∅ → P ∅ ) > h F ib (cid:0) A ∅ → holim( A → A , ← A ) (cid:1) ∪ h F ib( f )and finally get hFib( A ∅ → P ∅ ) > ΩΣΩ F > Ω F . (cid:3) ection 6. The Main Theorem P : ◻ → sSets and use thepairs of opposing faces ( ∂ , ∂ ¬ ), ( ∂ , ∂ ¬ ) and ( ∂ , ∂ ¬ ) of ◻ to build the corresponding strongpullback in several steps, yielding a sequence (6.1) of homotopy pullback cubes. Pictorially, weare going to do the following (using ( ∂ , ∂ ¬ ) here): P ∅ / / | | ②②②②②②② (cid:15) (cid:15) P (cid:15) (cid:15) z z ✈✈✈✈✈✈✈✈ P / / (cid:15) (cid:15) P , (cid:15) (cid:15) P / / | | ②②②②②②② P , z z ✈✈✈✈✈✈✈ P , / / P , , P (1)1 P (1) ∅ u u ❥❥❥❥ (cid:18) (cid:18) ✪✪✪✪✪✪✪ (cid:8) (cid:8) u u ❧❧❧ (cid:18) (cid:18) ✪✪✪✪✪ (cid:8) (cid:8) / / By definition, P (1) ∅ and P (1)1 are the homotopy pullbacks of ∂ ¬ P and ∂ P . Moreover, as de-scribed in section 1, by Thomason’s theorem and using the hypothesis that P is a homotopypullback cube, the square formed by P ∅ , P , P (1) ∅ and P (1)1 is also a homotopy pullback. Whenpassing to P (2) , we replace P (1) ∅ and P (1)2 by the homotopy pullbacks of ∂ ¬ P (1) and ∂ P (1) ,respectively. As the initial vertex changes, we need to make sure that the left face ∂ ¬ staysa homotopy pullback. Since everything is symmetric, we might just as well check that in thecubical diagram above, if the back and front faces are homotopy pullbacks, so is the squarecontaining P (1) ∅ , P (1)1 , P and P , . This follows from Thomason’s theorem, by which P (1) ∅ / / (cid:15) (cid:15) holim (cid:0) P → P , ← P (1)1 (cid:1) (cid:15) (cid:15) P / / holim (cid:0) P , → P , , ← P , (cid:1) is a homotopy pullback and the bottom map is a weak equivalence by assumption. Similarlywhen passing from P (2) to P (3) , where then, the cube has become the strong homotopy pullbackobtained from all the P ˆ k → P h i because all 2-faces of P (3) are homotopy pullbacks.(6.6) Proposition.
Let P : ◻ → sSets be a homotopy pullback cube and P ′ the newhomotopy pullback cube obtained from P by replacing P ∅ and P k with the homotopy pullbacksof ∂ ¬ k and ∂ k for some k ∈ h i . If all P M and P ′ M with M = ∅ are connected, then F P ′ l ⊆ F Pl (up to weak equivalences) for all l ∈ h i , whence F P ′ l ≫ F Pl . Proof.
Without loss of generality, let’s assume k = 1 (i.e. P ′ = P (1) ). The only vertices, where P and P (1) differ are those at ∅ and 1 and so, the only fibres that change are those of the morphismassociated to ∅ → F ) as well as those associated to ∅ → { l } and 1 → { , l } (in F l ) for1 = l . The first one is easy becausehFib (cid:16) P (1) ∅ → P (1)1 (cid:17) ≃ hFib (cid:0) P ∅ → P (cid:1) (because the corresponding square is a homotopy pullback) and therefore F P ′ ≃ F P . As for theother ones, we just use that P (1) ∅ and P (1)1 are obtained as the pullbacks of the left and rightfaces in the cube above. So, for example, taking l = 2,hFib (cid:16) P (1) ∅ → P (cid:17) ≃ hFib( P → P , ) , hFib (cid:16) P (1)1 → P , (cid:17) ≃ hFib( P , → P , , ) . So F P ′ comprises just these two fibres and is hence contained in F P . (cid:3) ection 6. The Main Theorem Proposition.
Let P , P ′ : ◻ → sSets and k ∈ h i as in the last proposition. If all P M and P ′ M with M = ∅ are connected, then the canonical transformation P ⇒ P ′ induces a maphocolim ⌜ P → hocolim ⌜ P ′ , whose homotopy fibre F satisfies F ≫ Σ (cid:0) F Pk ∗ hFib( P k → P ′ k ) (cid:1) .(6.8) Remark.
Implicit in our proposition is that hocolim ⌜ P ′ is connected as well, whichfollows readily from the Mayer-Vietoris long exact sequence for hocolim ⌜ P ′ . Proof.
Without loss of generality, let’s treat the case k = 1 (i.e. P ′ = P (1) ). Note that P ⇒ P ′ consists of identities everywhere except at ∅ and 1, where we have the homotopy pullback square P ∅ / / (cid:15) (cid:15) P (cid:15) (cid:15) P ′ ∅ / / P ′ . By the generalised Serre theorem (4.2), we get that the homotopy fibre F of the induced mapbetween homotopy colimits satisfies F ≫ Σ (cid:16) hFib( P ∅ → P ) | {z } ∈ F P ∗ hFib (cid:0) P ∅ → P ′ ∅ (cid:1)| {z } ≃ hFib( P → P ′ ) (cid:17) (cid:3) (6.9) Corollary.
Let P : ◻ → sSets be a homotopy pullback with corresponding stronghomotopy pullback S : ◻ → sSets and comparison map p : hocolim ⌜ P → hocolim ⌜ S . Ifall P M and S M with M = ∅ are connected, thenhFib( p ) ≫ Σ n F Pk ∗ hFib( P k → S k ) (cid:12)(cid:12)(cid:12) k ∈ h i o . Proof.
We construct S in several steps, as outlined in (6.1) and note that P (1)1 ≃ S , P (2)2 ≃ S and P (3)3 ≃ S , while P (1)2 ≃ P and P (2)3 ≃ P (1)3 ≃ P . Now, writing p k : hocolim ⌜ P ( k − → hocolim ⌜ P ( k ) (where P (0) := P )for the canonical maps, we have p = p ◦ p ◦ p . By the acyclic inequality (2.3) for composablepairs and (6.7), we havehFib( p ) > { hFib( p k ) | k ∈ h i} ≫ n Σ (cid:16) F P ( k − k ∗ hFib( P k → S k ) (cid:17) (cid:12)(cid:12)(cid:12) k ∈ h i o Using (6.6), we can now replace F P (1) and F P (2) by F P and F P , respectively. (cid:3) Finally, by combining this corollary with (6.5) and the square case (2.8), we obtain thefollowing weak version of the 3-dimensional homotopy excision theorem.(6.10)
Theorem.
Let A : ◻ → sSets be a strong homotopy pushout of connected spaces,with homotopy fibres F k := hFib( A ∅ → A k ) and comparison map q : A ∅ → holim ⌟ A . As longas the homotopy fibres F , F and F are again connected,hFib(Σ q ) > Σ(Ω F ∗ Ω F ∗ Ω F ) . eferences Proof.
Let P : ◻ → sSets be the cubical homotopy pullback associated to A and S : ◻ → sSets the strong homotopy pullback of all the A ˆ k → A h i . We first note that all P M and S M with M = ∅ are connected. Indeed, P → P (1)1 has no empty homotopy fibres becauseh F ib (cid:16) P → P (1)1 (cid:17) (2.8) > Ω hFib( P → P , ) ∗ Ω hFib( P → P , ) (2.7) > Ω F ∗ Ω F > S . Hence all components are hit and since P = A is connected, so is P (1)1 . Similarly whenpassing to P (2) and P (3) = S . Writing p : hocolim ⌜ P → hocolim ⌜ S , it suffices to show thathFib( p ) > Σ (Ω F ∗ Ω F ∗ Ω F ) because then, by (5.2),hFib(Σ q ) ≫ Ω hFib( p ) > ΩΣ (Ω F ∗ Ω F ∗ Ω F ) > Σ(Ω F ∗ Ω F ∗ Ω F ) . From the above corollary (and noting that P k = A k ), we already know thathFib( p ) ≫ Σ n F Pk ∗ hFib( A k → S k ) (cid:12)(cid:12)(cid:12) k ∈ h i o and by (6.5), we then havehFib( p ) > Σ { ΣΩ F k ∗ hFib( A k → S k ) | k ∈ h i} . But by definition S = holim( A , → A , , ← A , ), so that, by the acyclic homotopy excisiontheorem for squares (2.8) and Chachólski’s theorem (2.7),hFib( A → S ) > Ω hFib( A → A , ) ∗ Ω hFib( A → A , ) > Ω F ∗ Ω F . Similarly for the other two homotopy fibres, so that, all in all,hFib( p ) > Σ (Ω F ∗ Ω F ∗ Ω F ) . (cid:3) Combining this theorem with the relative Hurewicz isomorphism theorem [14, Theorem7.5.4], we recover a cubical version of the classical homotopy excision theorem [12] for simplyconnected spaces.(6.11)
Corollary. If A : ◻ → sSets is a strong homotopy pushout cube of simply connectedspaces whose homotopy fibres hFib( A ∅ → A k ) are i k -connected for i k >
1, then the total fibrehFib( q : A ∅ → holim ⌟ A ) is ( i + i + i )-connected. Proof.
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