Appendix Some Algebraically Topological Aspects in the Realm of Convenient Topology A.I Cohomology for filter spaces A .!.!. In order to give an exact definition of a cohomology theory for filter spaces we introduce at first the category Fih of pairs of filter spaces: Objects of F ih are pairs ((X,l'x) , (Y,I'Y)) - shortly (X, Y) - where (X,l'x) is a filter space and (Y, l'y) a subspace of (X, 'Yx) in Fil. Morphisms f : (X, Y) --+ (X', Y') are Cauchy continuous maps f : X --+ X' such that .f[Y] C Y ' . Defin it ion. Let G be a fixed abelian group. A cohomology theory for filter spaces with coefficients G is a pair (H* ,J*) where H* = (Hq)qE::Z is a family of contravariant functors H" : Fih --+ Ab from the category Fil2 into the construct Ab of abelian groups (and homomorphisms) for each intege r q and J* = (Jq)q E::Z is a family of natural transformations Jq : H" 0 T --+ Hq+l with a functor T : Fil2 --+ Fih defined by T(X,Y) = (Y,0) and T(.f) = fly for each f : (X, Y) --+ (X', Y') such that the following are satisfied : 1) Euaciness axiom. For any (X,Y) E IFil2 1 with inclusion maps i : (Y,0) --+ (X,0) and j : (X, 0) --+ (X, Y) there is an exact sequence ... c5Ell Hq(X, Y) ~ Hq(X, 0) ~ Hq(y, 0) c5~) Hq+l(X, Y) --+ ... 2) Uniform. homotopy axiom. If f : (X, Y) --+ (Z, W) and g : (X, Y) --+ (Z, W) are uniformly homotopic (i.e. if the pairs (X, Y) and (Z, W) of filter spaces are considered to be pairs of filtermerotopic spaces, there is a uniform homotopy H between f and g , i.e. a uniformly conti nuous map H : (X x I, Y x I) --+ (Z , W) such that H( ·,O) = f and H( ·,l) = g, where I denotes the unit interval [0,1] with its usual uniform structure and x stands for forming products in M er), 256 APPENDIX. ALGEBR.AICALLY T OPOLOG ICAL ASPECTS th en Hq(J) = Hq(g) for each integer q. 3) Excision axiom. IfY and U are subspaces of (X, ,) E IFill such that X \U E :F or Y E :F for each :F E" then the inclusion map i : (X\U,Y \ U) --+ (X,Y) induc es isomorphisms for each integer q. 4) Dimension axiom. If P is a filter space with a single point, th en 0 if f- 0 Hq(P, 0 ) ~ I q { G if q = O. A .1.2.1. From now on let G be a fixed abeli an group with more than one element. The Alexander cohomology group will be taken to be based on G and explicit denotation of G will be suppressed. Let (X, Y) E IFil2 1. Then the Alexander cochain complex is defined as follows: (1) For each non-negative integer q, let Fq(X) denote the abelian group of all functions f : Xn+I --+ G with the operations being defined pointwise, and Fq(X) = 0 for each negative integer q. A function f E Fq(X), q :::: 0, is called locally zero provid ed that for each Cauchy filter :F on X there is some F E :F such th at f (xo, . .. , xq) = 0 for any (q+ l j-tuple of elements of F. Th e subgroup of Fq(X) consisting of all locally zero functions is denoted by F6(X) , Let Cq(X) be the quotient group Fq(X)/F6(X), The inclusion map i : Y --+ X gives rise to a homomorphism Fq(i) : Fq(X) --+ Fq(y), defined by Fq(i)(J)(yo, .. ,Yq) = f(yo , . ,Yq), which indu ces a homomorphism Cq(i) : Cq(X) --+ Cq(y) , whose kernel is denot ed by Cq(X, Y). (2) Let us define homomorphisms 8q : Fq(X) --+ Fq+I(X), q :::: 0, by q+l q(J)( 8 xo, · · . ,Xq+l) = 2)-I)i.f(xo, . .. ,.ii,··· ,:Cq+l) i= O where Xi means th at Xi is to be omitted. 8q induces a homomorphism (oq) * Cq(X) --+ Cq+I(X ). Obviously, (8q)*[KerCq(i)] c KerCq+l(i). Th e induced homorphism ()q : Cq(X,Y) --+ Cq+I(X,Y) is called coboundary operator. Fur­ th ermore, ()q+l 0 ()q = 0 for each integer q. The quotient group Hq(X,Y) = Ker ()q / 1m ()q -l is called the q-dimensional Alexander cohomology group of the pair (X,Y) of filter spaces. In the following we write Hq(X) instead of Hq(X, 0). Obviously, Hq(X, X) = O. A .1.2.2. Let f : (X ,,) --+ (X', ,') be a Cauchy continuous map between filter A .1. COHOJ'vJ()L OGY FOR FILTER SPA.CES 257 spaces . Then a hom omorphism Fq(J ) : Fq(X') -+ Fq(X ) is defined as follows: Fq(J) = 0 if q < 0 Fq(J )(h) (xo, , xq) = h(J (xo), ... ,f(xq)) for each ne F q(X ) { and each (.'1:0, , xq) E x q+l if q 2: o. Since for each int eger q, Fq(J)[Fg(X' )] c Fg(X ), Fq(J ) indu ces a homomorphism c q(J) : c q(X ') -+ cq(X ). If (X, Y ), (X ', Y') E IFihl and f : (X ,Y) -+ (X ', Y') is Cauchy cont inuous, th en / c-(.f )[cq (X ', Y') ] c c q(X, Y ). Let us denote the restriction Cq(J)lcq(X ', Y ) again by Cq(J ). Then Cq(J ) : c q(X ', Y /) -+ Cq(X, Y) is a homomorphism which induces a homomorphism Hq(J) : Hq(X ', Y /) -+ H q(X , Y ). It is easily checked that H" : Fih -+ Ab is a contravariant functor, called th e q-dim ensional Alexander cohomology functo r- for- pair-s of filter- spaces. A. 1. 2.3. If (X, Y) E IFihl, th en th e homomorphism Cq(i) : Cq(X ) -+ Cq(y) induced by th e inclu sion map i : Y -+ X is surj ective (since Fq(i) : Fq(X) -+ Fq(y) is surjective for each int eger q). Now we can proceed as in t he to pological case (cf.[136; p. 308] or [76; Chapter 3, Section 5]) in defining th e connecting homomor-phism 6'1 : Hq(y) -+ H q+ l(X, Y). Furthermore, 6* = (6q)q E~ is a family of natural t ransformations 6'1 : H" 0 T -+ Hq+l, where the functor T is defined as und er 3.1. A.1.3. Exam p les. 1) Let (X ,r ) be a filter space and (X, J.loy) its corresponding filter-merotopic space . Then a function f E Fq(X ), q 2: 0, is locally zero in t he sense above iff it is locally zero in the merotopic sense, i.e. iff t here is some A E !loy such t hat for each A E A and each (q + 1)-tuple (xo, ... ,xq) of elements of A, f(xo, . , xq) = O. Furth ermore, subspaces in Fil-Mer are form ed as in Mer (cf. 7.1.3. 3) b)). Thus, if (X , Y) E jFihl is consider-ed to be a pair- of fi itermerotopi c spaces, the Alexander cohomology gTO Ups Hq(X , Y) as introduced above are nothing else than the Alexander cohomology gTO UpS of the merotopic pair' (X , Y) introduced in [11], which are isomorphic to the Gech cohomology groups iIlJ( X, Y) of the merotopic pair (X,Y) (cf. also [11]). 2) Let (X ,q) be a symmetric to pological space and (X ,rlJ ) its corres ponding filter space, i.e. r q = {F E F(X): th ere is some x E X with F ::> UlJ (x)} , T hen a function f E F q(X), q 2: 0, is locally zero in the sense above iff it is locally zero in th e usual topological sense, i.e. for each x E X th ere is some Ux E Uq(x ) such that for each (q + l j-tuple (xo, ... ,x q ) of elements of Ux , f (xo, ... , xq) = O. Since subspaces in Tops are formed as in KConvs and closed subspaces in KConvs are formed as in Fil (cf. 2.3.3.11. 3) and 2.3.3.27. 1)), a closed pair (X, Y) of symmetric topological spaces (i.e. X is a sym met ric topological space and Y a closed subspace of X) may be considered to be an element of IFihl. T hus, for closed pairs of symmetric topological spaces, the Alexander cohomology gTOUpS as defined above coincide with the Alexander cohomology gTO Up S in the 258 APPENDIX. ALGEBRAICALLY TOPOLOGICAL ASPECTS usual topological sense (cr. e.g. [76J or [136]). A.1.4 Theorem. If H" denotes the q-dimensional Alexander cohomologyfun c­ tor for pairs of filter spaces and 5'1 : Hq(y) --t Hq+l(X, Y) is the connecting ho­ momorphism, then (H*, 5*) is a cohomology theory for filter spaces provided that H* = (Hq)qE7L and 5* = W)qE7L . Proof It suffices to prove the four axioms for a cohomology theory (cf. A.I.I.). In this context it is useful to know that Bentley's variant of th e Eilenberg­ Steenrod axioms is valid for the Alexander cohomology of merotopi c pairs (cf. [11]). 1) It follows immediately from A.I.3. 1) that the exactness axiom is fulfilled, since it is fulfilled in the merotopic case. 2) The uniform homotopy axiom is fulfilled, since it its fulfilled in the merotopic case.