BLOCK STRUCTURES IN GEOMETRIC AND ALGEBRAIC TOPOLOGY

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1 Actes, Congrès intern. Math., Tome 2, p. 127 à 132. BLOCK STRUCTURES IN GEOMETRIC AND ALGEBRAIC TOPOLOGY by C. P. ROURKE I want to describe some homotopy functors which are defined on the category of CW complexes but which are naturally defined on the PL category first. The point being that the 'kinky' nature of PL topology turns out to be a positive advantage in the definition : we use the description of a polyhedron as an equivalence class of simplicial complexes under the relation of common subdivision. I shall give two examples of such functors and then a general recipe which includes both examples. Example 1. - "Block bundles". The first example is the classical example constructed by Kato [4], Morlet [7] and Rourke-Sanderson [10] ; it belongs in the realm of geometric topology. Let K be a PL cell complex, i.e. a polyhedron \K\ and a collection {a} of PL balls contained in \K\, the cells, which cover \K\ and satisfy (i) the interiors of the cells are disjoint (ii) the boundary of a cell or the intersection of two cells is a union of cells. A q-block bundle % q ' JK is a total space E ( ) D \K\ and for each /-cell a 6 K a block ß a C E(%) such that (ß a, a) is an unknotted (q + i, 0-ball pair. The blocks cover E(%) and satisfy axioms which can be summarised by saying that they "fit together like the cells of K"\ The crucial theorem for turning block bundles into a functor is the subdivision theorem : Let K' be a subdivision of K Then there is a natural 1: 1 correspondance between isomorphism classes of block bundles over K and block bundles over K'. The correspondence is established by subdividing the bundle itself, that is by finding blocks of Ç'/K* inside blocks of. In the next example I shall give a general proof of subdivision which includes this theorem. We can now define pullbacks for a PL map / : \L\ -> 1^1 by constructing the bundle x L/K x Lwith total space E( ) x \L\ and block ß a x r over a x r and then subdividing and restricting to r / C \K x L. And we get a homotopy functor as stated earlier (for details see [10 ; 1]). It is worth noting that there is also a natural notion of Whitney sum given by restricting x q/k x K to A K. Block bundles were invented to give a 'normal bundle' theory for the PL category.

2 128 C.P. ROURKE C 2 Example 2. "Cobordism". The second example belongs to algebraic topology. For details of this example, see Rourke-Sanderson [12]. A q-mock bundle % q /K consists of a total space 2s( ) and for each /-cell a G K a (q + /)-manifold M a C E(%), the blockovei a. The blocks cover ( ) and satisfy two axioms : o (i) M a are disjoint (ii) bm a = U {M T \rca} M a H M T = U im p p C a H r> Note the similarity of these axioms to those for a cell complex ; so we can again summarise the definition by saying that the "blocks fit together like the cells of K". «) K Figure 1 Picture of a 1-mock bundle ; the block over a is empty. Possible subdivisions corresponding to the new vertex ß are shown dotted. Subdivision theorem. Let K' be a subdivision of K and %/K a mock bundle. Then there is a mock bundle ÇIK' with E(Ç) = E( )andm a (?) = U{M T ( ') ircrf. In other words we cut up the blocks of f over cells of K'. As figure 1 illustrates, this theorem is a kind of transversality theorem ; so I am going to include a sketch of proof to stress the elementary nature of the method, which uses only collars. Sketch of proof By induction we can assume already subdivided over the (n \) skeleton of K and we have to extend over one «-cell a E K. If we can subdivide over a further subdivision o" of a', then on taking unions of blocks (amalgamating) we get a subdivision over &'. So we can assume that a 1 has a top dimensional cell a x C a and a' a x is a 'cylindrical triangulation', using a PL isomorphism of a' - a x with a' x /. Choose a collar on M a and define the blocks over cells of o' a x by identifying the two collar parameters, and finally define M a = M a -collar. Now subdivisions are not unique, but they are unique up to cobordism by the same proof, where mock bundles 0, % x are cobordant if they are restric-

3 GEOMETRIC AND ALGEBRAIC TOPOLOGY 129 tions of a mock bundle over K x I. And the cobordism classes of mock bundles define a contravariant functor T q ( ) on polyhedra. THEOREM. There is a natural equivalence T q ( )~&l- q ( ) where 2PL r PLdenotes the r-th unoriented PL cobordism group. Sketch of proof : It suffices to construct an Alexander duality isomorphism between mock bundles and bordism. As usual this follows from Poincaré duality. So let i Q /M n be a mock bundle over a manifold. Then E( ) is an (n + q) manifold (for proof see [3 ; 1-2]) and p : E(%) -> M (a projection constructed inductively over cells) is a bordism class which defines the Poincaré dual to. Conversely, given f : W -+ M then make / simplicial and consider dual cells in M. Then Z" -1 (cell) is a manifold by [2] and all the manifolds give a mock bundle structure with total space W. Finally to end this example we observe that the various operations in cobordism have easy geometric pictures in terms of mock bundles : Addition = disjoint union Cup product = Whitney sum (defined as for block bundles) Cap product = Amalgamated pull back : i.e. given f : W -+ K and %/k, form f*( )/W, then f.p: E(f*(%)) -> K is the required bordism class. General recipe. Let STI be a category of "pseudo-manifolds" and inclusions in the boundary, where a pseudo-manifold is an object with a virtual dimension and a boundary of one dimension lower. For a cell complex K define the associated category denoted K, as in [11 ; 1], to have objects the cells of K and morphisms the face inclusions in K. Then an (Oil, <?)-bundle, t q /K, is a functor : K-+VTC which raises dimension by q and such that the blocks (a) "fit together like the cells of K" i.e. (i) 3f(a)=u{ (r) rca} (ii) fw^fw = U{f(p) pgan r) Example (i) Ob (Oil) =id p xd q \p,q>0) dim (D p xd q )=p + q b(d p xd q ) = od p xd q (N.B. 3 not necessarily in OTd!)

4 130 C.P. ROURKE C 2 Mor (dît) = {/ : D p ' x D q UbD p x D q \ p* <p and f(d p ' x 0) C D' x 0} Then an (dïl, #>bundle is a #-block bundle. Ex (ii). (Block bundles with arbitrary fibre) Ob (OTc) = ip p xf} virtual dim. p d(d p xf)= dd p xf Mor (Oil) = {/ : D p ' x F^bD p x Fblockwise i.e. \m(f) = XxF, somex) Then an (dît, 0>bundle is a block bundle with fibre F in the sense of [1,11 ] with charts. Ex (iii). Homology cell analogue of (i) (Martin-Maunder [6]), where a homology cell is the cone on a homology manifold which is a homology sphere. This theory is the normal bundle theory for the "homology" category. Ex (iv). 31c = all manifolds (graded by dimension) then an (OH., #)-bundle is a #-mock bundle. Ex(v). 3ft = {manifolds with restriction on normal block bundle}. Then the corresponding mock bundle theory gives a more general cobordism theory. E.g. (a) normal bundle smooth oriented ; result smooth oriented cobordism. (b) normal bundle trivialised ; result stable cohomotopy. Another direction to generalise example (iv) is to introduce singularities. For example if we introduce all possible singularities : Ex (vi). dît = {principal w-polyhedra}, 8 = Z 2 -boundary. Then the resulting theory is Z 2 -cohomology (same proof as for mock bundles). Ex (vii). CTI = {Poincaré duality spaces}. Then the resulting theory is the cohomology theory corresponding to Levitt's "transversal subcomplex" of MG [5]. Axioms for a theory. We now axomatise the properties of dît which are needed to set up the theory : Axiom 1. Objects of dtl have collars up to cobordism. Axiom 2. -MG3R;=>Mx/earc. Axioms 1 and 2 allow the proof of the subdivision theorem to work, to provide subdivisions up to cobordism. Axiom 3 (amalgamation). Suppose M X,M 2,M X HM 2 E Oft, where M x andm 2 have "dim" n, andm x nilf 2 has "dim" n - 1, and that the inclusionsm x n Af 2 CM i3 i = 1, 2, are in Oïl. Then M x U M 2 G ZfC. Axiom 3 is necessary to pass from a bundle over K* to one over K by "amalgamating blocks",' Axioms 1, 2 and 3 imply independence of the cell structure of K and that we have a homotopy functor by the proof outlined for mock bundles.

5 GEOMETRIC AND ALGEBRAIC TOPOLOGY 131 Remark. There are significant cases where axioms 1 and 2 are satisfied but not axiom 3, for example OH, = unions of discs. In this case we can again define a functor by letting an object "over K" be an object over some subdivision of K. Amalgamation is then formal and the proof goes through. The functor corresponding to unions of discs, in codim 3 at least, is dual to "immersed bordism theory", for details see [14]. Axiom 4. - dît closed under disjoint union. Axiom 4 implies that we have an abelian semigroup functor under disjoint union ; in most natural cases, an abelian group. Axiom 5. - VTC closed under cartesian product. Axiom 5 gives an external product and by restriction an internal product (cup product, Whitney sum). Two final remarks ( 1 ) The general description of an Oil -bundle applies directly for a A-set, as in [11 ; 1 ], so that our functors are defined for the CW category. (2) There is a universal bundle y^/gsn, where G M is the "Grassmannian" of Oil-bundles over A ft embedded in A fc x 7? and 7^ is the obvious functor (compare [11 ; 1]). See also [8 ; 1]. Credits Sanderson and myself were awakened to the possibility of more general "block bundles" by the work of Martin and Maunder [6] ; however the construction has strong relations with the ideas of Casson and Sullivan, as exposited by myself [9], and, in a more general setting, by Quinn [8]. The terminology "mock bundle" is due to Cohen. REFERENCES [1] CASSON A. Block bundles with manifolds as fibres, (to appear). [2] COHEN M.M. Simplicial structures and transverse cellularity, Ann. of Math., 85 (1967), p [3] COHEN M.M. and SULLIVAN D.P. On the regular neighbourhood of a 2-sided submanifold, Topology, 9 (1970), p [4] KATO M. Combinatorial prebundles : I. [5] LEVITT N. Poincaré duality bordism, [6] MARTIN N. and MAUNDER C.R.F. Homology cobordism bundles, Topology (to appear). [7] MORLET C. Thesis. [8] QUINN. Thesis. [9] ROURKE C.P. Block bundles and Sullivan theory, (to appear). [10] ROURKE C.P. and SANDERSON B.J. Block bundles : I, Ann. of Math., 87 (1968), p

6 132 CP. ROURKE C 2 [11] ROURKE C.P. and SANDERSON B.J. A-sets : II, Quart. Jour. Math. (Oxford), (to appear). [12, 13, 14] ROURKE C.P. and SANDERSON B.J. Mock bundles I, II and HI, (to appear). University of Warwick Dept. of Mathematics, Coventry CV4 7AL Grande-Bretagne