Augmental Homology and the K"unneth Formula for Topological Joins
G"oran Fors
Department of Mathematics, University of Stockholm
S-106 91 Stockholm, Sweden *
Abstract. The "simplicial complexes" and"join"(*) today used within combina*
*torics isn't the classical con-
cepts, cf. [11] p. 108-9, but, except for ;, complexes having {;} as a subc*
*omplex resp. 1*2:={oe1[oe2|oei2i},
implying a tacit change of unit element w.r.t. the join operation, from ; *
*to {;}. Now, the classical
(co)homology theory automatically becomes obsolete, since it is inseparable*
* from its uniquely determined
source category. Obeying the Eilenberg-Steenrod formalism, this paper is de*
*dicated to the reconstruction,
and product/join exploration of the relative simplicial and singular homolo*
*gy theories in this new setting.
Explicitly;-H?(X;{;}) = H?(X), -H-1({;};;;G) = G, -H?(X;;) = eH?(X) and-H?(*
*X;Y ) = H?(X;Y ) else. Defining pair
joins through(X1;X2)*(Y1;Y2):=(X1*Y1;tX1*Y2[X2*Y1);he relative K"unneth for*
*mula for joins reads; (Th. 4p.6)
-Hq+1((X1;X2)*(Y1;Y2); G RG0) Re=i+j=q(-Hi(X1;X2; G) R-Hj(Y1;Y2; G0))i+j=*
*q-1TorR1(-Hi(X1;X2; G);H-j(Y1;Y2; G0)):
Complying with Whitehead's vocabulary below we'll call the --H-functor theA*
*ugmentalHomologyFunctor.
1 INTRODUCTION
Classical Simplicial and Singular Homology Theories have always been accomp*
*anied by the
Reduced Homology Functor in a mix governed only by the the skill of the individ*
*ual mathematician.
When, within Combinatorics, the classical category of simplicial complexes K we*
*re abandoned some
25 years ago in favor of the contemporary {;}-extended category of simplicial c*
*omplexesiKot would
have been quite natural, as in p. 3, to define a Simplicial Homology Theory for*
*malizingathisnmixd
through the realization functor p. 3 also to define an associated Singular Homo*
*logy Theory, as in p. 4.
The join operation is extremely important within algebraic topology. For in*
*stance it is fundamen-
tal to Milnor's construction of the universal principal fibre bundle in [8] whe*
*re he also formulate the
non-relative K"unneth formula for joins as; eHq+1(X1* Y1) Ze==i+j=q(eHi(X1) ReH*
*j(Y1))i+j=q-1TorZ1(eHi(X1);ieHj(Y1)):e:
the "X2=Y2=;"-case in our Th. 4 p. 6. These results, apparently, inspired G.W. *
*Whitehead to introduce
the Augmental Total Chain Complex eS(O) and Augmental Homology,He?(O); in [12].
Whitehead's introduction do differ, for reasons now given, from that of our*
*s. G.W. Whitehead
states that eS(X * Y ) and eS(X)eS(Y ) are chain equivalent,, cf. our Theorem 3*
* p. 6 and then in a
footnote he points out, referring to [8] p. 431 Lemma 2.1, that; -"This fact do*
*es not seem to be stated
explicitly in the literature but is not difficult to deduce from Milnor's proof*
* of the "K"unneth theorem"
for ... join...". But, X * ; =X, [12] p. 56, and -"...He0(X,;) is the reduced *
*(0)-dimensional homology
group of X ", [12] p. 57, implying eSi(;) 0 8i2 Z and so; eS(X) = eS(X * ;) e*
*S(X) eS(;) 0:
The above contradiction steams from the fact that G.W. Whitehead gave the e*
*mpty space, ;;
the status of a (-1)-dimensional standard simplex but never took into account t*
*hat ; then would
get the identity map, Id;, as a generator for its (-1)-dimensional singular aug*
*mental chain group.
On the chain level it's indeed"not difficult"to see what's needed to achiev*
*e eS(X*Y)eS(X)Se(Y);
but then to actually do it for the classical category of topological spaces and*
* within the frames of the
Eilenberg-Steenrod formalism is, unfortunately, impossible, since the need for *
*a (-1)-dimensional stan-
dard simplex is indisputable and the initial object ; just won't do. Instead we*
*'ll use a familiar routine
from Homotopy Theory providing free spaces with a common base point, i.e. we di*
*sjointly add to each
topological space an element " and choose {"} to be our desired (-1)-dimensiona*
*l standard simplex
and join unit. To this collection we add ; without which really nothing would h*
*ave been achieved.
G.W. Whitehead has been one of the most distinguished typologists of all ti*
*mes and the rest of
[12] is, of course, reliable, for instance (2.3) p. 57, the "X2= ; or Y2= ;"-ca*
*se of our Theorem 4 p: 6:
Whitehead statement, almost 50 years ago, that no proof was stated explicitly i*
*n the literaturefor
eS(X * Y) eS(X) eS(Y) has been true ever since except that our Theorem 3 p. 6 *
*now providesasuch
proof. Whitehead uses a "built in" dimension shift in eS(X) while we're using a*
* suspension operator s.
_________________________
* E-mail; goranf@matematik.su.se
1
2. AUGMENTAL HOMOLOGY THEORY
2.1. DEFINITIONS OF UNDERLYING CATEGORIES AND NOTATIONS
Let K be the classical category of simplicial complexes and simplicial *
*maps, with vertices
belonging to a common universe W. The typical morphisms of K are the simplicial*
* maps, as de-
fined in [11] p. 109, which, in particular, implies; MorK (;; ) = {;} = {0;; };*
* MorK (; ;) = ; if
6= ; andMorK (;; ;) = {;} = {0;;;} = {id;} where 0;0 = ; = the empty function*
* from to0:
So; 0;0 2 MorK (; 0) () = ;: If in a category we have 'i2 Mor (Ri; Si) (i = 1;*
* 2) we define
ae
'1t '2 : R1t R2 -! S1t S2 : r 7! '1(r)' if r 2 R1; where " t " := "disjoi*
*nt union".
2(r)if r 2 R2
Definition of the objects in Ko: An (abstract) simplicial complex on a vertex *
*set V is a collection
(empty or non-empty) of finite (empty or non-empty) subsets oe of V satisfying;
(a) If v 2 V , then {v} 2 . (b) If oe 2 and o oe then o 2 .
So, {;} is allowed as an object in Ko. We will write "concepto" or "concept*
*"" when to stress that
a concept relates to our extended categories. If |oe| = #oe := card(oe) = q+1 t*
*hen dimoe := q and oe is
said to be a q-faceoor a q-simplexo of o and dimo:=sup{dim(oe)|oe 2 o}. Writing*
* ;owhen using ;
as a simplex, we get dim(;)=-1 and dim({;o})=dim(;o)=-1:
Note that each object in the category Koof simplicial complexeso except for ;, *
*includes {;o} as
a subcomplex. A typical objectoin Kois t {;o} or ; where 2 K and is a morph*
*ism in Koif;
(a) = ' t id{;}for some' 2 MorK (; 0) or
(b) = 0;;o: (In particular; MorKo (o; {;}) = ; if and only if o 6= {;}; ;*
*:)
A functor E: Ko-!K:
Set E(o) = o\ {;} 2 Obj(K) and given a morphism :o ! 0owe put;
E( ) = ' if fulfills (a) above and
E( ) = 0;;E(o)if fulfills (b) above.
A functor Eo: K -!Ko:
Set Eo()= t {;}2Obj(Ko)and given simplicial ': ! 0 ; putting := ' t id{;}, gi*
*ves;
_ EEo =idK
_ imEo = Obj(Ko) \ {;}
_ EoE = idKexcept for EoE(;) = {;}
Similarly, let C be the category of topological spaces and continuous maps.*
* Consider the category
D"with objects: ; together with X":=X +{"}, for allX 2 Obj(C); i.e. the set X":*
*=X t{"} equipped
with the weak topology, oX", with respect to X and {"}, cf. [3] Def. 8.4 p. 132.
8
< a) f+ id{"}(:= f t id{"}) withf 2 MorC(X; Y ) andf *
*isonX__toY; i:e
f"2 MorD"(X"; Y") if; f"= : the domain off is thewhole ofX and X"= X+ {"}; Y"= *
*Y +{"} or
b) 0;;Y"(= ; = the empty function from; toY"):
D" -! C
There are functors F":X C7-!!D"X+;{"} F:X+ {"} 7! X resembling Eoresp. E.
; 7! ;
Note. The "F"-lift topologies", oXo:=oX[ {Xo}={Oo|Oo=Xo\ (N t {"}) ; N closed i*
*nX} [ {Xo} and
oXo:=F"(oX) [ {;}={Oo=O t {"} | O 2oX} [ {;} would also give D"due to the domai*
*n restriction in a;
making D" a link between the two constructions of partial maps referred to in [*
*1] pp. 184-6.
No extra morphisms" has been allowed into D"(Ko) in the sense that the morphism*
*s" are all pic-
tures under F"(Eo) except 0;;Y"defined through item b, re-establishing ; as the*
* unique initial object.
2
NOTATIONS: We have used w.r.t.:=with respect to, and oX:=the topology of X. We'*
*ll also use;
PID := Principal Ideal Domain, l.h.s.(r.h.s.):=left (right) hand side, iff :=if*
* and only if, cp.:=compare
(cf. = cp.!), LHS:=Long -Homology Sequence and M-Vs:=Mayer-Vietoris sequence. L*
*et = {p;@} be
the classical singular chain complex and let "'" denote "homeomorphism" or "cha*
*in isomorphism".
Let o (oo) denote the one (two) point space {o; "} ({oo; "}).
If X"6= ;; {"} then (X"; oX") is a non-connected space, and it therefore se*
*ems adequate to define
X"6= ;; {"} to have a certain point set topological "property"" if F(X") has th*
*e "property" in question.
For instance; X"is connected"if and only if X is connected.
2.2. SIMPLICIAL AUGMENTAL HOMOLOGY THEORY AND REALIZATIONS
We will let -Hdenote the simplicial as well as the singular augmental (co)h*
*omology functoro.
Choose oriented q-simplices to generate Coq(o; G), where the coefficient mo*
*dule G is a unital
($1AOg = g) module over any commutative ring A with unit.
Co(;; G) is identically 0 in all dimensions, where 0 denotes the additive unit-*
*element.
Co({;o};G) is identically 0 in all dimensions except in dimension -1 where Co-1*
*({;o};G) "=G:
Co(o; G) eC(E(o); G) "{;o}-augmented chain" as defined in ordinary algebraic *
*topology.
By just hanging on to the "{;o}-augmented chains", also when defining relat*
*ive chainso, we get
the Relative Simplicial Augmental Homology Functor for Ko-pairs, denoted -H*and*
* fulfilling;
8
>>>Hi(E(o1); E(o2); G)if o26= ;
< eHi(E(o1);aG)e if o16= {;o}; ;, and o2= ;
-Hi(o1; o2; G) = > =eG if i = -1
>>:= 0 if i 6= -1 when o1={;o} and o2= ;
0 for all i when o1= o2= ;.
;6={;o} and both lacks final sub-objects, which under any useful definition*
* of the realization of a
simplicial complex implies that |;| 6= |{;o}| demanding the addition of a non-*
*final object {"} = |{;o}|
into the classical category of topological spaces as join-unit and (-1)-dimensi*
*onal standard simplex.
This approach conforms Homology Theory and considerably simplifies the study of*
* manifolds, cf. p. 8.
We will use Spanier's definition of the "function space realization" |o| as*
* given in [11] p. 110,
unaltered, except for the " o"'s and the underlined addition__where " := ff0 an*
*d ff0(v) 0 8 v 2 V :
---" We now define a covariant functor from the category of simplicial comp*
*lexeso and simplicial
mapso to the category of topological spaceso and continuous mapso. Given a non*
*empty simplicial
complexo o , let |o|be the set of all functions ff from the set of vertices of *
*o to I := [0; 1] such that;
(a) For any ff, {v 2 Vo |ff(v) 6= 0} is a simplexo of o (in particular, ff(v) *
*6= 0 for only a finite set
of vertices). P
(b) For any ff 6=_ff0__, v2Vff(v) = 1:
o
If o = ; , we define |o|= ;. " q ___________
P *
* 2
The barycentric coordinateso ff, defines a metric d(ff; fi) = [ff(v)-fi(*
*v)]; on |o| inducing the
v2Vo
topological space |o|d with the metric topology. We'll equip |o| with another t*
*opology and for this
purpose we define the closed simplexo |oeo| of oeo 2 o i.e. |oeo| := {ff 2 |o| *
*| [ff(v) 6= 0] =) [v 2 oeo]}:
Definition. For o 6= ;, |o| is topologized through |o|:= |E(o)| + {ff0}, which*
* is equivalent to
give |o| the weak topology w.r.t. the |oeo|'s, naturally imbedded in Rn+{"} and*
* we define o to be
connected if |o| is, i.e. if F(|o|) ' |E(o)| is: (o|o|= o|o|diff_o is locally f*
*inite by [11] p. 119 Th. 8.)
Proposition. ([5] pp. 115, 226) The function space realizationo|o| is homotopy *
*equivalent to|o|d: __|_|
3
2.3. SINGULAR AUGMENTAL HOMOLOGY THEORY
|oe"| imbedded in Rn+{"} generates a satisfying set of "standard simpl*
*ices"" and "singular
simplices"". This implies in particular that the "p-standard simplices"", deno*
*ted"p , are defined by
"p :=p +{"} wherep denotes the usual p-dimensional standard simplex and + is th*
*e topological
sum, i.e."p :=p t {"} with the weak topology w.r.t.p and {"}. In particular:"(*
*-1):= {"}.
Let T pdenote an classical, arbitrary singular p-simplex (p 0). The "sin*
*gular p-simplex"",
denoted oe"p, now stands for a function of the following kind:
oe"p:"p =p +{"} -! X+ {"} where oe"p(") = " and
oe"p|p= T pfor some ordinaryp -dimensional singular simplexT pfor all*
*p 0:
In particular; oe"(-1): {"} -! X"= X+ {"} : " 7! ":
The boundary function @" is defined by @"p(oe"p) := F"(@p(T p)) ifp > 0 w*
*here @p is the ordinary
singular boundary function, and @"0(oe"0) oe"(-1)for every singular 0-simplex*
*"oe"0: Let" = {"p; @"}
denote the singular augmental chain complex": Observation;|o| 6= ; =)|o| = Fff*
*0(|E(o)|) 2 Dff0:
By the strong analogy to classical singular homology, H, we omit the proof*
* of the next lemma:
Lemma. (Analogously for coH-omology.)
8
>>>Hi(F(X"1); F(X"2); G)if X"26= ;
> e=G if i = -1
>>> = 0 if i 6= -1 when X"1={"} and X"2= ;
: 0 for all i when X *
* __
"1=X"2= ;. *
* |_|
Note. i.i(X1; X2; G)e="i(F"(X1); F"(X2); G) always. In particular, -H-1(F"(X1)*
*; F"(X2); G) 0:
ii." (X"1; X"2) '(F(X"1); F(X"2)) except iff_X"16= X"2= ; when the only non-i*
*somorphisms
occur for"(-1)(X"1; ;) e=Z 6=e0 e=(-1)(F(X"1); ;) and -H0(X"1; ;) Z e=H0(F*
*(X"1); ;) if X"16={;}.
iii.Co(o1; o2; G)" (|o1|; |o2|; G) (see p. 1 for )connects the simplicial and*
* singular functoro:
iv.-H0(X"+oY"; {"}; G) = -H0(X"; {"}; G) -H0(Y"; {"}; G) but -H0(X"+oY"; ;; *
*G) = -H0(X"; ;; G)
-H0(Y"; {"}; G)= -H0(X"; {"}; G) -H0(Y"; ;; G) with X"+oY":=F"(F(X")+F(Y")*
*) if X"6=;6=Y":
Definition: The "pth Singular Augmental Homology Group of X" w.r.t. G":= -Hp(X*
*"; ;; G): The
Coefficient Groupo:= -H-1({"}; ;; G): Using F"(Eo); we "lift" the concepts of *
*homotopy, excision and
point in C (K) into D"-concepts (Ko-concepts) homotopy", excision" and point" *
*(=: o), respectively.
So; fo; go 2 D" are homotopic" if and onlyfifo= go = 0;;Yoor there are hom*
*otopic maps
f1; g1 2 C such that fo = f1+ Id{"}; go = g1+ Id{"}:
An inclusiono (io; ioAo) : (Xo\ Uo; Ao\ Uo) -! (Xo; Ao) is an excision" if*
* and only if there is an
excision (i; iA) : (X \ U; A \ U) -! (X; A) such that io = i + Id{"}and ioAo= *
*iA + Id{"}:
{P; "} 2 D"is a point"iff_{P} + {"} = F"({P}) and {P}2 C is a point. So, {*
*"} is not a point":
Conclusion: (-H; @"), abbreviated -H, is a homology theory on the h-category o*
*f pairs from D" (Ko),
c.f. [4] p. 117, i.e. H- fulfills the h-category analogues, given in [4] xx8-9*
* pp. 114-118, of the seven
Eilenberg-Steenrod axioms from [4] x3 pp. 10-13. The necessary verifications a*
*re either equivalent to
the classical or completely trivial. E.g. the dimension axiom is fulfilled sin*
*ce {"} is not a point":
Since the exactness of the relative Mayer-Vietoris sequence of a proper tr*
*iad, follows from the
axioms, cf. [4] p. 43 and, paying proper attention to Note iv, we'll use it wi*
*thout further motivation:
eH(X)=H-(F"(X); ;)explaines most of the ad-hoc reasoning surrounding the c*
*lassical eH-functor.
4
3 3:1 AUGMENTAL HOMOLOGY MODULES FOR JOINS AND PRODUCTS
Let r be one of the classical topological product/join operations "x"// " **
* "// "^*", defined in [11]
p. 4 // in [3] p. 128 Ex. 3 including [3] p. 135 Problem 6:1, [9] p. 373 and [*
*12] p. 128 // in [1] pp.
159-160 and [9] resp. Recall;aX*e; =X = ; * X classically. We will let stand *
*for "[" or "\".
Definition. X"1roX"2:= ;F if X"1= ; or X"2= ;:
"(F(X"1)rF(X"2) ) if X"16= ; 6= X"2
From now on we'll delete the "/o-indices. So, e.g. "X connected" now means *
*"F(X) connected":
Equivalent Join Definition. Put ; t1X =Xt1;:=;. If X6=;; {"}t1X ={"}, Xt1{"}=X.*
* For X;Y 6=
;; " let Xt1Y denote the set X xY x(0; 1] pasted to the set X by '1: X xY x{1} *
*-! X; (x; y; 1) 7! x ;
i.e. the quotient set of X x Y x (0; 1] t X; under the equivalence relation *
*(x; y ; 1) ~ x and let
p1: X xY x(0; 1] tX -! Xt1Y be the quotient function. For X; Y = ; or{"} let X*
*t0Y := Y t1X and
else the set X x Y x [0; 1) pasted to the set Y by the function '2: X x Y x {0}*
* -! Y ; (x; y; 0) 7! y ;
and let p2: Xx Y x [0; 1) t Y -! Xt0Y be the quotient function. Put XOY := Xt*
*1Y [ Xt0Y :
(x; y; t) 2 XxY x [0; 1] specifies the point (x; y; t) 2 Xt1Y \ Xt0Y; one-t*
*o-one, if0 < t < 1 and
the equivalence class containing x if t = 1 (y if t = 0), which we denote (x; 1*
*) ((y ; 0)): This allows
"coordinate functions" : XOY ! [0; 1], j1: Xt1Y ! X; j2: Xt0Y ! Y extendable t*
*oXOY through
j1(y; 0):x02X resp. j2(x; 1):y02Y and a projection p: X t X x Y x [0; 1] tY !XO*
*Y:
Let X^*Y denote XOY equipped with the smallest topology making ; j1; j2 con*
*tinuous andX*Y;
XOY with the quotient topology w.r.t. p, i.e. the largest topology makingp cont*
*inuous()oX^*YoX*Y):
Pair-definitions: (X1; X2)r (Y1; Y2):=(X1roY1; (X1roY2) (X2roY1)); where, if e*
*ither X2or Y2 is
not closed cf. [3] p. 122 Th. 2.1(1), (X1* Y1; (X1* Y2) (X2* Y1)) has to be*
* interpreted as
(X1* Y1; (X1O*Y2)(X2O*Y1)) i.e. (X1OY2)(X2OY1) with the subspace topology in th*
*e 2:nd component.
Analogously for simplicial complexes with "x" ("*") from [4] p. 67 Def. 8.8 ([1*
*1] p. 109 Ex. 7, cf. p. 1).
Note: (X1; {"}) x (Y1; Y2) = (X1; ;) x (Y1; Y1) ifY2 6= ; and (X1O Y2) \ (X2O *
*Y1) = X2O Y2.
(Xt1Y )t0:5in X*Y is homeomorphic to the mapping cylinder w.r.t. the coordinate*
* map q1:XxY !X:
X2^*Y2is always a subspace of X1^*Y1by [1] 5.7.3 p. 163. X2*Y2is a subspace of *
*X1*Y1if X2,Y2are closed.
^*is associative by [1] p. 161 and * is claimed (non-)associative in ([1] p. 16*
*2) [9] p. 373 Ex. 4.
* and ^*are both commutative. "x\" is (still, cf. [2] p. 15) the categorical pr*
*oduct on pairs from D".
Through Lemma+Note ii p. 4 we convert the classical K"unneth formula (line *
*4), cf. [11] p. 235,
mimicking what Milnor did, partially ( line 1), at the end of his proof of [8] *
*p: 431 Lemma 2:1:
Theorem 1. For {X1xY2; X2xY1} excisive, q 0, R a PID, and assuming TorR1(G; G0*
*)=0 then;
-Hq((X1; X2) x (Y1; Y2); G RG0)=Re
8 0 0 0
>><[-Hi(X1; G) RH-j(Y1; G )]q(-Hq(X1; G) RG ) (G R-Hq(Y1; G )) T1 if C andX*
*2=;=Y2
R [-Hi(X1; G) R -Hj(Y1; Y2; G0)]q (GR -Hq(Y1; Y2; G0)) T2 ifC andX2=;6*
*=Y2
=e>>[-Hi(X1; X2; G) -Hj(Y1; G0)]q (-Hq(X1; X2; G) G0) T3 ifC andX26=;=Y*
*2(1)
: [-H R 0 R
i(X1; X2; G) R -Hj(Y1; Y2; G )]q T4 ifX1xY1=;;{"} orX2 6= ; 6*
*= Y2
The T-terms splits as those ahead of them, resp., e.g. T4= [Tor1R(-Hi(X1; X2; G*
*); -Hj(Y1; Y2; G0))]q-1
and T1 = [Tor1R-Hi(X1; G);H-j(Y1; G0) ]q-1 Tor1RH-q-1(X1; G); G0 Tor1RG ;H-q*
*-1(Y1; G0) : L
C :="X1xY16= ;; {"}" and [...]q is still, i.e. as in [11] p. 235 Th. 10, to be *
*interpreted asi+j=...q&i;j0:_|_|
Lemma. Let f : (X; A) ! (Y; B) be a relative homeomorphism, i.e., f : X ! Y is *
*continuous and
f :X\A! Y \B is a homeomorphism. If F:NxI!N is a strong (neighborhood) deformat*
*ion retraction
of N down onto A and B and f(N) are closed in N0:=f(N\A)[B;atheneB is a strong *
*(neighborhood) de-
formation retract of N0through; F 0: N0xI ! N0; (y,t)(7!yy,t)f7!OifFy(2fB,-t12*
*(Iy),it)f y 2 f(N)\B = f(N\A), t 2 I.
Proof. F 0is continuous as being so when restrictedto f(N)xI resp. BxI; cf. [1]*
* p. 34; 2:5:12: __|_|
5
Theorem 2. (Analogously for ^*instead of *:) If (X1; X2)6=({"}; ;)6=(Y1; Y2) a*
*nd G an A-module;
-Hq((X1; X2)x(Y1; Y2); G) Ae=-Hq+1((X1;X2)*(Y1;Y2); G)H-q((X1;X2)*(Y1;Y2)t0:5+*
*(X1;X2)*(Y1;Y2)t0:5;G) =
8
>>-Hq+1(X1* Y1; G) H-q(X1; G) H-q(Y1; G)if X1xY16= ;; {"} andX2= ; =Y2
A< -Hq+1((X1; ;) * (Y1; Y2); G) H-q(Y1;iY2;fG)X1xY16= ;; {"} andX2= ; 6=Y2
=e> *
*(2)
>:-Hq+1((X1; X2) * (Y1; ;); G) H-q(X1;iX2;fG)X1xY16= ;; {"} andX26= ; =Y2
-Hq+1((X1; X2) * (Y1; Y2); G) if X1xY1= ;; {"} or X26= ; 6=Y2
Proof. Split X*Y at t=0:5 then; X (Y ) is a strong deformation retract of (X*Y*
* )t0:5((X*Y )t0:5): The
relative M-Vs w.r.t.the excisive couple of pairs {(X1;X2) * (Y1;Y2)t0:5; (X1;X*
*2) * (Y1;Y2)t0:5} splits since the
inclusion of their topological sum into (X1;X2)*(Y1;Y2) is pair null-homotopic*
*, cf. [9] p. 141 Ex. 6c, and
since the 1:st(2:nd) pair is acyclic if Y2(X2)6=; we get Theorem 2:Equivalentl*
*y for ^*by the Lemma. __|_|
5.7.4 [1] p. 164: There is a homeomorphism: : X^*Y ^*E0! (X^*E0)x(Y ^*E0) whi*
*ch restricts to a
homeomorphism: X^*Y ! (X^*E0) x Y [ X x (Y ^*E0): (Here E0 is a symbol for a p*
*oint.) __|_|
Corollary 5.7.9 [7] p. 210: If OE: C E with inverse and OE0: C0 E0with inve*
*rse 0, then
OE OE0: C C0 E E0with inverse 0. *
* __|_|
Theorem 46.2 [9] p. 279: For free chain complexes C; D vanishing below a certa*
*in dimension and if a
chain map : C !D induces homology isomorphisms in all dimensions, then is a c*
*hain equivalence._|_|
Theorem 3. (The relative Eilenberg-Zilber theorem for joins.) For an excisive *
*couple {X^*Y2; X2^*Y}
from the category of ordered pairs ((X; X2); (Y; Y2)) of topological pairs"; s*
*(" (X; X2) " (Y; Y2)) is
naturally chain equivalent to" ((X; X2) ^*(Y; Y2)): ("s" stands for suspensio*
*n i.e. the suspended chain
equals the original except that dimension i in the originalis dimension i+1 in*
* the suspended chain.)
Proof. The second isomorphism is the key and is induced by the pair homeomorph*
*ism in [1] 5.7.4 p.
164. For the 2:nd last isomorphism we use [7] p. 210 Corollary 5.7.9 and that *
*LHS-homomorphisms
are "chain map"-induced. Note that the second component in the third module is*
* an exicive union.
-Hq(X^*Y)=Z'-Hq+1(X^*Y^*{v;Z"};=X^*Y)'-Hq+1((X^*{u;"})x(Y ^*{v; "}); ((X*^{u;"*
*})xY)[(Xx(Y ^*{v; "})))=
Motivation: The underlying chains*
* on the l.h.s. and r.h.s. are,
= -Hq+1((X^*{u; "}; X)x(Y ^*{v; "}; Y )) Z=' bywNotehiiip.c4,hisomorphicwtoet*
*heiruclassicalscounterpartseontheZclassical=Eilenberg-Zilber'Theorem.
Z " ^ " ^ Z " " Z " "
='-Hq+1( (X* {u; "}; X)Z (Y *{v; "};HYq))='+1(s (X)Zs (Y))='Hq(s[ (X)Z:(Y*
*)])
Now the non-relative Eilenberg-Zilber Theorem for joins follows from [9] p*
*. 279 Th. 46.2. .
Substituting, in the x-original proof [11] p:234; " ^*", "" /s" ", "Th. 3,*
* 1:st part" for "x", "",
"Theorem 6" resp. will do since; s" (X1) " (Y1) = s" (X1) " (Y2) + s" (X2*
*) " (Y1) =
" " " " " " " " " *
* " __
=s (X1) (Y1)= (X1) (Y2) + (X2) (Y1) = s (X1)= (X2) (Y*
*1)= (Y2) : |_|
[11] Cor. 4 p. 231 now gives Th. 4 since; -H?(O) "=s-H?+1(O) "=-H?+1(s(O))*
* and" ((X; X2) * (Y; Y2))
" ((X; X2)^*(Y; Y2)) by Th. 2. and [9] p. 279 Th. 46.2. ({X1*Y2; X2*Y1} is exc*
*isive iff_{X1xY2;)X2xY1} is.
Theorem 4. (cp. [11] p. 235.) If {X1^*Y2; X2^*Y1} is an excisive couple in X1^*
**Y1; R a PID ; G; G0
R-modules and TorR1(G; G0) = 0, then the functorial sequences below are (non-n*
*aturally) split exact;
L
0 -! i+j=q[-Hi(X1; X2; G) R-Hj(Y1; Y2; G0)] -! *
* (3)
L
-!H-q+1((X1; X2) ^*(Y1; Y2); G RG0)- !i+j=q-1TorR1-Hi(X1; X2; G);H-j(Y1; Y2; G*
*0) -! 0
Analogously with " * " substituted for "^*" and [11] p. 247 Th. 11 gives t*
*he co-Homology-analog. __|_|
Theorem 5. [The Universal Coefficient Theorem for (co)-Homology.] ( Put (X1; X*
*2)=({"}; ;) in Th: 4:)
R R R
H-i(Y1; Y2; G) Re=[11] p: 214e=-Hi(Y1; Y2; R RG) e=H-i(Y1; Y2; R) RG Tor1-Hi*
*-1(Y1; Y2; R); G ;
for any R-PID module G. If all -H*(Y1; Y2; R) are of finite type or G is finit*
*ely generated, then;
-Hi(Y1; Y2; G) Re=-Hi(Y1; Y2; R RG) Re=-Hi(Y1; Y2; R) RG TorR1-Hi+1(Y1; Y*
*2; R); G : __|_|
6
3.2. LOCAL AUGMENTAL HOMOLOGY GROUPS WITH RESPECT TO PRODUCTS AND JOINS
Lemma. x (y) closed inX (Y ) =) {Xx(Y \ y); (X \ x)xY }; {X* (Y \ y); (X \ x) **
* Y } both excisive:
Proof. [11] p. 188 Th. 3 since X x(Y \ y0) X *(Y \ y0) is open in (X x(Y \ y0*
*)) [ ((X \ x0) xY )
(X*(Y \y0))[((X\x0)*Y ) which, here, is equivalent to being open in XxY X*Y *
*, cf. [3] p. 122: __|_|
Our definition of a `setminus", "\o", in D" is motivated by Proposition 1 b*
*elow and reflects the
non-local i.e. the non-geometrical nature of the (-1)-element ".
ae; if X = ;, X X0 or X0 = {"}
Definition. X"\oX0":= F 0 " "66=" "
" F(X" ) \ F(X") else
Proposition 1 is our key motivation for introducing a topological (-1)-obje*
*ct. "X \ x" usually
stands for "X \ {x}" and we will write x for {x; "} as a notational convention.*
* See p. 3 for Def. of ff0.
Proposition 1. Let G be any module over a commutative ring A with unit. With ff*
* 2 Intoe and ff = ff0
iff_oe =;othe following module isomorphisms are all induced by chain equivalenc*
*es; cf. Th. 46.2:p. 6
A A A
-Hi-#oe(Lk oe;eG)=-Hi(; costoe; G) e=-Hi(||; |costoe|; G) e=-Hi(||; ||*
* \off; G)
A i A i A i
-Hi-#oe(Lk oe; G) e=-H(; costoe; G) e=-H(||; |costoe|; G) e=-H(||; || *
*\off; G):
Proof. (Cf. "More definitions" p. 8.) The " \o"-definition above, [9] Th. 46.*
*2 p. 279 + pp. 194-
199 Lemma 35.1-35.2 +Lemma 63.1 p. 374 gives the two ending isomorphisms since *
*|cost_oe|_is a
deformation retract of ||\off; whilealready on the chain level; Co?(; costoe) =*
* Co?(st(oe); _oe*Lk oe)=
= Co?(__oe* Lkoe; _oe* Lkoe) '?Co-#oe(Lk oe): (This result is, somewhat special*
*ized, found in [6] p. 162 and
partially also in [10] p. 116 Lemma 3.3.) *
* __|_|
Proposition 2. If x2X (y 2Y ) closedand (t1; gx*y; t2):={(x; y; t) | 0 < t1tt2<*
*1};
i. H-q+1(X ^*Y; X ^*Y \o(x; y; t); G) Ae==-Hq+1(X ^*Y; X ^*Y \o(t1; gx *;yt2); *
*G) Ae==-Hq(XxY; XxY \o(x; y); G)=Ae=
A A simple A A Th: 2 p: 6 A *
* ^
==e [calculation] e==-Hq((X; X\ox)x(Y; Y \oy);eG)==[ line four] e==-Hq+1(*
*(X; X\ox) *(Y; Y \oy); G):
ii. H-q+1(X*^Y; X*^Y \o(y; 0); G) Ae==-Hq+1((X; ;) ^*(Y; Y \oy ); G) and equiva*
*lently for the (x; 1)-points.
All isomorphisms are induced by chain equivalences, cf. p. 6. Analogously with"*
* * "substituted for"^*":
ae 1 ae ^
Proof. i. AB:=:X=tYX\o{(x0;ty0;0t)Y|\t1ot{<(1}x=) A [ B =X* Y \o(t1; gx *;yt*
*2)
0 ; y0;At)\|B0=