MIT OpenCourseWare http://ocw.mit.edu 18.917 Topics in Algebraic Topology: The Sullivan Conjecture Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Atomicity of Connected p-Finite Spaces (Lecture 29) In this lecture, we will show that every connected p-finite space K is an atomic object of the category of p-profinite spaces. We begin with the following result, which is a soft version of the convergence of the homology spectral sequence for a cosimplicial space: Proposition 1. Consider the functor F : X �→ C∗X from the category of p-profinite spaces S∨ to the category of complexes of F -vector spaces. Then F carries p p totalizations (of cosimplicial p-profinite spaces) to geometric realizations (of simplicial objects in the category of complexes). Proof. The functor F factors as a composition F �� ◦ F �, where: • The functor F � is given by the formula X �→ C∗X, but C∗X is regarded as an E -algebra over F . ∞ p • The functor F �� is the forgetful functor from E -algebras over F to complexes of F -vector spaces. ∞ p p We have shown that the functor F � carries arbitrary homotopy limits of p-profinite spaces to homotopy colimits of E -algebras; in particular, it carries totalizations to geometric realizations. It now suffices to ∞ observe that F �� preserves geometric realizations of simplicial objects. Remark 2. In concrete terms, the statement that F �� preserves geometric realizations amounts to the following observation. Let A be a simplicial object in the category of E -algebras over F , and let |A | • ∞ p • be the geometric realization of A as a complex of F -vector spaces. Then |A | inherits the structure of • p • an E -algebra. For example, the multiplication on |A | arises in the following way: the tensor product ∞ • |A |⊗ |A | can be identified with the homotopy colimit of the bisimplicial complex B given by the formula • • •,• B = A ⊗ A . This homotopy colimit can be computed as the geometric realization of the diagonal m,n m n simplicial object B :[n] �→ B , and we have a map • n,n |B |→|A | • • which is induced by the maps B = B = A ⊗ A → A given by the multiplication on A . n n,n n n n n Theorem 3. Let S∨ denote the category of p-profinite spaces, and c S∨ the category of cosimplicial p­ p p profinite spaces. Then the totalization functor c S∨ → S∨ p p commutes with homotopy pushouts. 1 Proof. Suppose given a homotopy pushout diagram �� X• X�• � � � � �� Y • Y �• of p-profinite spaces. We wish to show that the associated diagram Tot(X•) ��Tot(X�•) � � � � Tot(Y •) ��Tot(Y �•) is again a homotopy pushout diagram. In other words, we wish to show that the canonical map Tot(X�•) � Tot(Y •) → Tot(Y �•) Tot(X•) is an equivalence of p-profinite spaces. As we saw in the last lecture, it will suffice to show that this map induces an isomorphism on cohomology. In other words, it suffices to show that the induced map C∗(Tot(Y �•)) → C∗(Tot(X�•) � Tot(Y •)) Tot(X•) is a quasi-isomorphism. By excision, the right hand side can be identified with the homotopy fiber product (C∗ Tot X�•) × (C∗ Tot Y •). C∗ Tot X• In other words, we must show that the diagram C∗ To� �t(X•) �� C∗ Tot� �(X�•) C∗ Tot(Y •) �� C∗ Tot(Y �•) is a homotopy pullback square (in the category of complexes of F -vector spaces). Using Proposition 1 we p can rewrite this square as |C∗X� � •| �� |C∗X� � �•| |C∗Y •| �� |C∗Y �•|. The homotopy theory of complexes of F -vector spaces is stable: that is, homotopy pullback squares p are the same as homotopy pushout squares. It will therefore suffice to show that the diagram above is a homotopy pushout square. Since the collection of homotopy pushout squares is stable under homotopy colimits (and in particular under geometric realizations), we are reduced to showing that each diagram �� C∗X� � n C∗X� � �n �� C∗Y n C∗Y �n 2 is a homotopy pushout square. Using stability again, we need only show that this diagram is a homotopy pullback square. This follows by excision from the assumption that �� Xn X�n � � � � �� Y n Y �n is a homotopy pushout square of p-profinite spaces. Remark 4. The same argument can be used to show that the formation of filtered limits of p-profinite spaces preserves homotopy pushout squares. Corollary 5. Let K be a simplicial object in the category of p-finite spaces. Suppose that each K is atomic • n in the p-profinite category, and that the geometric realization K = |K | is again p-finite. Then K is also • atomic in the p-profinite category. Proof. Let X be any p-profinite space. Since each Kn is atomic, we can construct a mapping object XKn . The universal property of XKn shows that it depends on Kn in a contravariantly functorial manner, so that F (X) = XK• is a cosimplicial object in the category of p-profinite spaces, and the totalization of this cosimplicial object can be identified with XK . Consequently, the functor X �→ XK can be factored as a composition S∨ →F c S∨ T→ot S∨. p p p Since each K is atomic, the functor F preserves homotopy pushouts. Theorem 3 implies that Tot preserves n homotopy pushouts as well, so the functor X �→ XK preserves homotopy pushouts. Since K is clearly nonempty, we deduce that K is atomic. Corollary 6. Let G be a finite p-group and n ≥ 1 an integer; assume that G is abelian if n> 1. Then the Eilenberg-MacLane space K(G,n) is an atomic object of the p-profinite category. Proof. The case n = 1 was handled in the previous lecture, using Lannes’ T-functor. The proof in general goes by induction on n. Choose a fibration E → K(G,n), where E is contractible, and consider the 0 0 associated simplicial object defined by the formula E = E × E × ... × E k 0 K(G,n) 0 K(G,n) 0 (where the factor E appears (k + 1)-times). Then E is a simplicial object whose geometric realization |E | 0 • • can be identified with K(G,n). Moreover, each E is homotopy equivalent to the Eilenberg-MacLane space k K(Gk,n − 1). The inductive hypothesis implies that each E is atomic. Using Corollary 5, we conclude that k |E |� K(G,n) is atomic as well. • Theorem 7. Let X be a connected p-finite space. Then X is an atomic object of the p-profinite category. Proof. The space X admits a Postnikov tower X � X → X → ... → X �∗, m m−1 0 where � π X if k ¡ i π X = k k i 0 otherwise. 3 We will show by induction that each X is atomic. For each i, we have a fiber sequence i K(π X,i) → X → X i i i−1 of connected p-finite spaces. Consequently, to show that X is atomic, it will suffice to show that X and i i−1 K(π X,i) are atomic. The first follows from the inductive hypothesis and the second from Corollary 6. i We conclude this lecture with a complement to Theorem 7, which indicates the strength of the atomicity condition. Proposition 8. Let X be an atomic object in the category S of spaces. Then X is (weakly) contractible. Proof. By assumption Map(X, •) commutes with finite colimits. In particular, Map(X, ∅) is empty; this im­ plies that X is nonempty. Since Map(X, •) commutes with disjoint unions, we conclude that X is connected. Suppose (for a contradiction) that X is not weakly contractible. Without loss of generality, we can assume that X is a CW complex. It can therefore be written as the homotopy colimit of its skeleta sk0 X → sk1 X → sk2 X → ... Let X� denote the mapping telescope for this sequence of maps, so that we have a canonical homotopy equivalence X� � X. The telescope X� admits a decomposition � X� X� 1 2 X� 0 where: • The space X� is the disjoint union of the spaces ski X. 0 • The space X� is the disjoint union of the mapping cylinders for the inclusions ski X ⊆ ski+1 X, where 1 i is odd. • The space X� is the disjoint union of the mapping cylinders for the inclusions ski X ⊆ ski+1 X, where 2 i is even. Since X is atomic, the equivalence X � X� factors (up to homotopy) through either X� or X�. Since X 1 2 is connected, we this map factors through the mapping cylinder of the inclusion ski X ⊆ ski+1 X, for some integer i. Consequently, we deduce that the identity map from X to itself factors up to homotopy through the finite dimensional spaceski+1 X. We now prove, by induction descending induction on j, that the identity map id factors (up to homo­ X topy) through skj X. The case j = i + 1 follows from the above argument. For the inductive step, we use the homotopy pushout diagram � Sj−1 ��skj−1 X � D� � j ��skj � �X. Since X is atomic, we conclude that id factors (up to homotopy) either through skj−1 X or through �Dj . X Since X is connected, the latter possibility implies that id factors up to homotopy through some disk Dj , X which contradicts our assumption that X is not contractible. Therefore id factors through skj−1 . X Applying the above argument repeatedly, we deduce that id factors through the sk−1 X = ∅. Since X X is nonempty, we obtain a contradiction. Where does the above argument go wrong if we work in the p-profinite category? 4