intrinsic strong shape for topological spaces
TRANSCRIPT
INTRINSIC STRONG SHAPE FOR TOPOLOGICAL SPACES
NIKITA SHEKUTKOVSKI, BETI ANDONOVIC
Abstract. In this paper is presented an intrinsic definition of strong shape for
topological spaces. At first a coherent proximate net YXf : is defined, indexed by
finite subsets of normal coverings of Y, and then there is a homotopy between two coherent
proximate nets defined. A definition of composition of classes of homotopies between two
coherent proximate nets YXf : and ZYg : is given. Then it is proved that for other
choice of corresponding coverings, a function is obtained that is in the same class with the
previously defined composition. The strong shape category is obtained, with strong paracompacta as objects, and the homotopy classes of coherent proximate nets as morphisms.
Key words: Coherent proximate net, covering, homotopy class, intrinsic, strong shape
0. Introduction
The shape theory is more appropriate tool than homotopy in study of spaces
with bad local behavior. The strong shape theory is a strengthening of shape theory.
The first definition of strong shape for compact metric spaces is given in [3]
by embedding metric compacta in Hilbert cube. In [2], [5] strong shape is described
for topological spaces approximating the space by inverse system of polyhedra
(ANRs). Both approaches follow the corresponding approaches in shape theory.
The intrinsic approach to shape does not use any approximation of spaces.
The intrinsic definition of strong shape for compact metric spaces is presented in [4].
1. Coherent proximate nets
By a function YXf : we mean a mapping from X to Y . f is not
necessarily continuous.
Let Y be a strongly paracompact space. We define the following sets:
}finitestaris,ofcoveringais|{)(* UUU YYCov .
Let }max,),(|{)( *
max
* aaYCovaaYCov . In )(max
* YCov we have the
following ordering: baba . If ba , then ba maxmax .
Let }0...1|),...,,{(, 2121 nn
nnn ttttttR be the non standard
n -simplex.
Definition 1.1. A coherent proximate net YXf : is a set of functions
},...,0),(|...),,...,,(|{ max
*010 niYCovaaaaaaaff inna
such that each YXf n
a : continuousmaxis 0аstn and is )(max 0
1 astn -
continuous on Xn , and the following coherence condition is satisfied
0),,,...,(
),,,...,ˆ,...,(
1),,,...,(
),,...,,(
11...
11...ˆ...
12...
21
10
0
1
nnаа
iiniааа
nаа
nа
txttf
ttxtttf
txttf
xtttf
n
ni
n
Special cases.
If 0n , for each 0a , there exists YXfa :0
, so that 0af is 0max a -
continuous.
If 1n , for each 0a , there exists YXfa :0
, so that 0af is 0max a -
continuous and for every ,, 10 aa there exists ,: 1
10YXf aa such that
10aaf is
)(max 0аst - continuous and af is 0max a - continuous on X1 , and
).(),1(),(),0(110010
xfxfxfxf aaaaaa
The coherent proximate net, CPNf , we will shortly denote by )( aff .
Definition 1.2. Let YXgf :, be CPNs. We say f and g are homotopic
(denotation: gf ), if there exists a coherent proximate net ,: YXIH
)( aHH , such that ,: YXIH n
а is )(max 0
1 аstn -continuous, аH is
)(max 0аstn -continuous on ,)( XIn and the following conditions are satisfied:
).,(),1,(
),,(),0,(
xtgxtH
xtfxtH
аа
аа
The relation gf is an equivalence relation. This is shown in [7].
Remark 1.3. The equivalence relation at CPN homotopies does not
necessarily hold if we drop the requirement including the condition of the homotopies
being stnmaxa-continuous, and substitute it by the condition of the homotopies being
just amax -continuous. There was an example given by Akaike and Sakai [6]
showing that in such case, the relation of homotopy might not meet the transitive
property requirement, and therefore not be an equivalence relation.
2. Composition of coherent proximate nets – existence and uniqueness
The following two Theorems we need for the definition of composition of
coherent proximate nets. Theorem 2.2 is proved in [Nsh TopProc] for the case k = 1.
Theorem 2.1. If YXf : is W-continuous, then YKXKfid : is
WK -continuous, where K is compact and }|{ WW WWKK .
Proof. YXf : is W-continuous, and it follows ,Xx U -neighborhood
of x and WW , so that WUf )( .
As usually fid is defined by ))(,(),( xfkxkfid . Then, for
XKxk ),( , there exists U , U a neighborhood of x and there exists
WKWK , so that WKUKfid )( .
Theorem 2.2. Let W be a star finite covering of Z and ZYG k: be a
)(Wkst - continuous function and )(1 Wkst - continuous on Yk . Then there exists
V - a star-finite covering of Y , so that for each function YXf : that is V -
continuous, ZXfidG k:)( is )(Wkst - continuous, and )( fidG is
)(1 Wkst - continuous on Xk .
Proof. Let Yy be a fixed point. For each kks \ , there exists kk
sJ \ , a neighborhood of s and a neighborhood s
yV of y , so that
s
y
s
ys WVJG *)( , for some element )(* Wks
ystW .
For each ks , there exists NJ s , a neighborhood of s and a
neighborhood s
yV of y , so that s
y
s
ys WVJG )( , for some element )(1 Wks
y stW .
Then }|{ k
s sJ is an open covering of k
. There exists a finite subcovering
1sJ ,
2sJ ,...psJ .
We put ps
y
s
yy VVV ...1.
Then i
i
s
yys WVJG *)( , for kk
is \ , and i
i
s
yys WVJG )( , for
k
is .
}|{ YyVy is a covering of Y . Since Y is strongly paracompact, there exists a
finer covering V that is star-finite. Let VV . There exists y , so that yVV . Then
the following holds: i
i
s
ys WVJG *)( , for kk
is \ , and i
i
s
ys WVJG )( , for
k
is .
},,...1|{ VVpiVJis is a star-finite covering of Yk
. If YXf : is
a V - continuous function, then there exists a star finite covering U of X , so that
VU)(f .
Now, if we define ZXH k: by:
_
))(,(),( xftGxtH ,
then H is a )(Wkst - continuous function and H is )(1 Wkst - continuous on
Xk .
We will now define a partitioning of the simplex
}0...1|),...,,{( 2121 nn
n tttttt
by defining the sets ,0, nmK m in the following way:
}2
1|),...,{( 11 mmnm ttttK .
Now, let )(max
* YCovB and )(max
* ZCovC , and let YXff b :)( and
ZYgg c :)( be CPNs. In order to define the composition ZXhh c :)( we
will perform an induction by the height of the element Cc .
Definition 2.3 Let A be an ordered cofinite set. Then the height )(ah of
Aa is defined as follows:
}...|max{)( 110 aaaanah n .
We define a mapping BCg : , such as follows:
Case 0. Let 0)(, chCc .
We choose an element Bb , so that cbstg max)max( . Then we
put bcg )( .
Now, we define ZXhc : by )(cgcc fgh . Then ch is max c –continuous.
Case 1. Let 1)(, chCc .
We define )(cg , choosing Bb , so that the following conditions hold:
1. cbstg max)max( ;
2. bcg )( 0 for all possible choices of 0c , cc0
3. )(0 bcc fidg is cst max - continuous and )(
0 bcc fidg is cmax -
continuous on X1 ;
We put bcg )( .
We define the mappings ch and cch0
as follows:
)(cgcc fgh .
Then ch is max c –continuous.
We define the mapping ZXh cc
1:0
by:
1
0
),,2(
)),(,12(),(
00
0
0 Ktxtfg
Ktxftgxth
bbc
bcc
cc
Because of Theorem 2.1, bbc fg00
is 0max cst - continuous, and because of
Theorem 2.2 and the condition 3 , )(0 bcc fidg is 0max cst - continuous and
)(0 bcc fidg is 0max c - continuous on X1 ;
Then cch0
is 0max cst -continuous and cch0
is 0max c - continuous on X1 .
At the picture 1 below there is a given review of the mapping cch0
.
bbc fg00
bcc fg
0
|______________|______________|__ t
0 0К 2
1 1К 1
Picture 1
Case n-1 (inductive assumption).
We assume that for each c having a height ,1)( nch there is defined bcg )( , so
that
1) cbstg max)max( ;
2) ,)()(...)()( 210 bcgcgcgcg n for all possible choices of indices
cccc n 210 ... ;
3) The following mappings are defined: cccccccccc nhhhh
210100 ......,,,, ,
20 ni , such that ZXh i
ccc i
1
... :0
is 0
1 maxcst i - continuous and ccc ih ...0
is
0max cst i - continuous on Xi 1 .
At the picture below there is a given review of the mapping ccch10
.
Picture 2
2t
2
1
1К 0К
2К
1t
bbbc fg100
bccc fg10
bbcc fg110
1. cbstg max)max( ;
2. bcg )( 0 , for all possible choices of 0c , cc0 ;
3. )(0 bcc fidg is cst max - continuous and )(
0 bcc fidg is cmax -
continuous on X1 .
Case n. Let nchCc )(, .
Here it is important to mention that there exists a linear homeomorphism
between the sets iK and ini mapping vertices to vertices.
We choose b , so that:
1. cbstg max)max( ;
2. bcg )'( , for all possible choices of 'c , cc' ;
3. ZXfidg knk
bbcc nkk:)( ...... 10
is cst k max -continuous, for
nk ,...,2,1 and for bmax , )( ...... 10 nkk bbcc fidg is cstk max1 - continuous on
Xknk .
Then we put bcg )( .
Because of the inductive assumption, here we have:
,)()(...)()( 110 bcgcgcgcg n for each ccccc nn 1210 ... .
Also, there are defined the following mappings: cccccccccc nhhhh
210100 ......,,,, .
We define the mapping ZXh n
cccc n:
110 ... by :
2
1)),(,12,...,12(
2
1)),,2,...,2(,12,...,12,12(
2
1),,2,...,2,2(
),(
1...
11...21...
121...
...
110
110
1100
110
nbncccc
iinibbbbicc
nbbbbc
cccc
txfttg
ttxttftttg
txtttfg
xth
n
niii
n
n
Then, because of Theorem 2.1, Theorem 2.2 and the condition 3, cccc nh
110 ... is
0max cst n - continuous and cccc nh
110 ... is 0
1 max cst n - continuous on Xn .
We check that cccc nh
110 ... is well defined:
Let .2
11t
).,2,...,2(
),2,...,2,1(),2,...,2,2
12(
2...
2...2...
110
11001100
xttfg
xttfgxttfg
nbbbc
nbbbbcnbbbbc
n
nn
).,2,...,2()),2,...,2((
)),2,...,2(,0(),2,...,2(,12
12(
2...2...
2...2...
110110
11101110
xttfgxttfg
xttfgxttfg
nbbbcnbbbc
nbbbccnbbbcc
nn
nn
Let .2
11ii tt From 1ii tt ,
)).,2,...,2(,12,...,12,12(
)),2,...,2(,12,...,12,12(
1...121...
1...21...
1110
110
xttftttg
xttftttg
nibbbbicc
nibbbbicc
niii
niii
From 2
1it ,
)),2,...,2(,12
12,...,12,12(
)),2,...,2(,12,...,12,12(
1...21...
1...21...
110
110
xttfttg
xttftttg
nibbbbcc
nibbbbicc
niii
niii
)).,2,...,2(,12,...,12,12(
)),2,...,2(,0,...,12,12(
1...121...
1...21...
1110
110
xttftttg
xttfttg
nibbbbicc
nibbbbcc
niii
niii
Let .2
1nt
)).(,12,...,12())(,0,12,...,12(
))(,12
12,...,12())(,12,...,12(
11...11...
1...1...
110110
110110
xfttgxfttg
xftgxfttg
bncccbncccc
bccccbncccc
nn
nn
)),
2
12(,12,...,12,12(
)),2,...,2(,12,...,12,12(
110
110
121...
1...21...
xftttg
xttftttg
bbncc
nibbbbicc
nn
niii
)).(,12,...,12,12(
)),1(,12,...,12,12(
121...
121...
10
110
xftttg
xftttg
bncc
bbncc
n
nn
It follows that the function is well defined.
We obtain that the composition of the coherent proximate nets
YXff b :)( and ZYgg c :)( is a coherent proximate net ZYhh c :)( .
We denote this composition by YXfg : .
In order to conclude that the composition is well defined, the next thing to
prove is that the following composition of classes of coherent proximate nets is well
defined:
][][][ fgfg .
In fact, it is enough to show that for another choice of Bb , such that it
satisfies the same required conditions 1, 2, 3, and 4, using the above definition of
composition of coherent proximate nets, there is obtained a function of composition
that is in the same homotopy class.
We use the set ,...}2,1,{ iaA i , which is ordered cofinite set, and the height
of Aa :
}...|max{)( 110 aaaanah n .
We have defined a composition of YXf : and ZYg : by induction of
the height.
We use the following denotations again:
)(max
* YCovB , )(max
* ZCovC , and the defined partitioning of the non-
standard simplex }0...1|),...,,{( 2121 nn
n tttttt by the sets ,mK
.0 nm
Case 0.
Let 0)(, chCc .
We have defined a mapping BCg : , in the following way: For Cc , we
chose Bb , so that cstbstg max)max( .
We put bcg )( .
Then we defined ZXhc : by )(cgcc fgh .
Then ch is max c –continuous .
Now, for each c , let us use another element with the same properties as b ,
and let that element be 'b . So we have )(' cgb .
In the same way as above, we define the mapping '
ch in the following way:
'
'
bcc fgh , where '
ch is max c – continuous.
We chose '," bbb , where "b satisfies the same conditions as b , i.e.
cbstg max)"max( , and for each cc0 , ")( 0 bcg . We define the mapping ch"
in the following way:
"" bcc fgh , where ch" is max c – continuous.
Then, we define ZXIH c : by ),(),( " xtfgxtH bbcc .
By that we obtain a homotopy between bcc fgh and "" bcc fgh .
Case 1.
Let 1)(, chCc .
We have defined )(cg , choosing Bb , so that:
1. cbstg max)max( ;
2. For )()(: ** ZCovYCovg , ccbcg 00 ,)( .
3. For bmax , )(0 bcc fidg is cst max - continuous and )(
0 bcc fidg is
cmax - continuous on X1 .
We have put bcg )( .
We define the mappings ch and cch0
in the following way:
)(cgcc fgh , ch is max c –continuous.
The mapping ZXh cc
1:0
we define by:
0
1
),,2(
)),(,12(),(
00
0
0 Ktxtfg
Ktxftgxth
bbc
bcc
cc
In such a case, cch0
is 0max cst -continuous and cch0
is 0max c - continuous on
X1 .
Let us now, for each c and cc0 , choose another element with the same
properties as b , and let that element be 'b . So we have ''
0 bb , where )( 0
'
0 cgb and
)(' cgb .
Same as above, we define the mappings '
ch and '
0cch in the following way:
'
'
bcc fgh , where '
ch is also max c –continuous. The mapping
ZXh cc
1' :0
we define by:
0
1'
),,2(
)),(,12(),(
''00
'0
0 Ktxtfg
Ktxftgxth
bbc
bcc
cc
'
0cch is 0max cst -continuous and '
0cch is 0max c - continuous on X1 ,
because of Theorem 2.1, Theorem 2.2 and condition 3 above.
There exists '," bbb and ""0 bb , so that '
000 ," bbb , and "b satisfies the
same conditions as b . We define the mappings ch" and cch0
" in the following way:
"" bcc fgh , where ch" is also max c –continuous.
The mapping ZXh cc
1:"0
we define by:
0""
1"
),,2(
)),(,12(),("
00
0
0 Ktxtfg
Ktxftgxth
bbc
bcc
cc
cch0
" is 0max cst -continuous and cch0
" is 0max c - continuous on X1 ,
because of Theorem 2.1, Theorem 2.2 and condition 3 above.
We define a homotopy ZXIH c : by:
),(),( " xsfgxsH bbcc .
By that we show that bcc fgh is homotopic to "" bcc fgh .
It is left for us to show that cch0
is homotopic to cch0
" .
We define ZXIH cc
1:0
in the following way (picture 3):
Picture 3
12
1)),,(,12(
2
1
2),,,2(
20),,2,(
),,(
"
"
""
0
00
000
0
txsftg
ts
xstfg
stxtsfg
xstH
bbcc
bbbc
bbbc
cc
Let us show that ccH0
is well defined on the edges:
- For 2
st ,
),(),,(),2
2,( """"" 00000000xsfgxssfgx
ssfg bbcbbbcbbbc .
t
s
"0000 bbcc fgH
"bbcc fgH
""00 bbc fg
"0 bcc fg
bcc fg0
bbc fg00
""000 bbbc fg
"00 bbbc fg
"00 bbc fg
"0 bbc fg
"0 bbcc fg
cch0
"
cch0
2
1
On the other hand,
),(),,(),,2
2( """ 000000xsfgxssfgxs
sfg bbcbbbcbbbc
.
- For 2
1t ,
),(),,1(),,2
12( """ 00000
xsfgxsfgxsfg bbcbbbcbbbc .
On the other hand,
),()),(,0()),(,12
12( """ 000
xsfgxsfgxsfg bbcbbccbbcc.
The following also holds:
- For 0s ,
12
1)),,0(,12(
2
10),,0,2(
0),,2,0(
),0,(
"
"
""
0
00
000
0
txftg
txtfg
txtfg
xtH
bbcc
bbbc
bbbc
cc
12
1)),,0(,12(
2
10),,0,2(
0),,0,0(
"
"
""
0
00
000
txftg
txtfg
txfg
bbcc
bbbc
bbbc
12
1)),(,12(
2
10),,2(
0),,0(
0
00
000 "
txftg
txtfg
txfg
bcc
bbc
bbc
12
1)),(,12(
2
10),,2(
0),(
0
00
00
txftg
txtfg
txfg
bcc
bbc
bc
),(
12
1)),(,12(
2
10),,2(
0
0
00
xth
txftg
txtfg
cc
bcc
bbc
.
- For 1s ,
12
1)),,1(,12(
2
1),,1,2(
2
10),,2,1(
),1,(
"
"
""
0
00
000
0
txftg
txtfg
txtfg
xtH
bbcc
bbbc
bbbc
cc
12
1)),,1(,12(
2
1),,1,1(
2
10),,2,1(
"
"
""
0
00
000
txftg
txfg
txtfg
bbcc
bbbc
bbbc
12
1)),(,12(
2
1),,1(
2
10),,2,1(
"
"
""
0
0
000
txftg
txfg
txtfg
bcc
bbc
bbbc
12
1)),(,12(
2
1),(
2
10),,2(
"
"
""
0
0
00
txftg
txfg
txtfg
bcc
bc
bbc
),("
12
1)),(,12(
2
10),,2(
0
0
00
"
""
xth
txftg
txtfg
cc
bcc
bbc
.
By that we show that ccH0
is a homotopy between cch0
and cch0
" .
Case n.
We have defined:
2
1)),(,12,...,12(
2
1)),,2,...,2(,12,...,12,12(
2
1),,2,...,2,2(
),(
1...
11...21...
121...
...
110
110
1100
110
nbncccc
iinibbbbicc
nbbbbc
cccc
txfttg
ttxttftttg
txtttfg
xth
n
niii
n
n
and similarly as in the previous case, the following is defined:
2
1)),(,12,...,12(
2
1)),,2,...,2(,12,...,12,12(
2
1),,2,...,2,2(
),("
"1...
11""...""21...
121""...""
...
110
110
1100
110
nbncccc
iinibbbbicc
nbbbbc
cccc
txfttg
ttxttftttg
txtttfg
xth
n
niii
n
n
We define a homotopy ZXIH n
cccc n:
110 ... in the following way:
We have defined the partitioning of the non-standard simplex
}0...1|),...,,{( 2121 nn
n tttttt by the sets ,0, niK i where
}2
1|),...,{( 11 iini ttttK . Now we need a partitioning of the sets IK i , so we
define the sets ,j
iK for each inj ,...,1,0 in the following way:
}.2
10,,,
2
1
2,,
thatso,),,...,(|),,...,,...,{( 11
lm
inni
j
i
tnljnlts
jnmim
IKsttstttK
Now we define ZXIH n
cccc n:
110 ... on IK i , ni ,...,1,0 , by:
)),,2,...,2,,...,2(,12,...,12(),,,...,( 11"...""...1...1... 110110xttstfttgxsttH njnibbbbbicccncccc jnjnjniin
for j
in Kstt ),,...,( 1 .
We will show that cccc nH
110 ... is well defined. In fact, we need to observe
several cases for which we will show that cccc nH
110 ... is well defined:
1. For j
i
j
in KKxstt 11 ),,,...,( , ni ,...,1 , we will show that the definition of
cccc nH
110 ... is unique.
2. For j
i
j
in KKxstt 1
1 ),,,...,( , inj ,...,1 , we will show that the
definition of cccc nH
110 ... is unique.
3. We will show that the definition of cccc nH
110 ... on the edges of In
coincides to the corresponding homotopies on In 1, i.e., to ccccc ni
H110 ...ˆ... , ni ,...,1 .
1. Let j
i
j
in KKxstt 11 ),,,...,( , i.e. 2
1it .
On one hand, because of j
in Kxstt 11 ),,,...,( and 2
1it , we have:
)),2,...,2,,...,2(,12,...,12(),,,...,( 1"...""...11...1... 11100xttstfttgxsttH njnibbbbbiccncc jnjnjnii
)),2,...,2,,...,2
12(,12,...,12(),,,...,( 1"...""...11...1... 11100
xttsfttgxsttH njnbbbbbiccncc jnjnjnii
)),2,...,2,,...,1(,12,...,12(),,,...,( 1"...""...11...1... 11100xttsfttgxsttH njnbbbbbiccncc jnjnjnii
)),2,...,2,,...,2(,12,...,12(),,,...,( 11"...""...11...1... 1100xttstfttgxsttH njnibbbbbiccncc jnjnjnii
On the other hand, because of j
in Kxstt ),,,...,( 1 and
2
1it , we have:
)),2,...,2,,...,2(,12
12,...,12(),,,...,( 11"...""...1...1... 100
xttstftgxsttH njnibbbbbccncc jnjnjnii
)),2,...,2,,...,2(,0,...,12(),,,...,( 11"...""...1...1... 100xttstftgxsttH njnibbbbbccncc jnjnjnii
)),2,...,2,,...,2(,12,...,12(),,,...,( 11"...""...11...1... 1100xttstfttgxsttH njnibbbbbiccncc jnjnjnii
by which we show case 1.
2. Let j
i
j
in KKxstt 1
1 ),,,...,( , i.e. 2
1
st jn
.
On one hand, because of 1
1 ),,,...,( j
in Kxstt and 2
1
st jn
, we have:
)),2,...,2,,2
2,...,2(,12,...,12(),,,...,( 21"..."...1...1... 1100xtts
stfttgxsttH njnibbbbiccncc jnjnii
)),2,...,2,,,...,2(,12,...,12(),,,...,( 21"..."...1...1... 1100xttsstfttgxsttH njnibbbbiccncc jnjnii
)),2,...,2,,2,...,2(,12,...,12(),,,...,( 21"..."...1...1... 100xttsttfttgxsttH njnjnibbbbiccncc jnjnii
On the other hand, because of j
in Kxstt ),,,...,( 1 and 2
1
st jn
, we have:
)),2,...,2,2
2,,...,2(,12,...,12(),,,...,( 21"...""...1...1... 100xtt
sstfttgxsttH njnibbbbbiccncc jnjnjnii
)),2,...,2,,,...,2(,12,...,12(),,,...,( 21"...""...1...1... 100xttsstfttgxsttH njnibbbbbiccncc jnjnjnii
)),2,...,2,,2,...,2(,12,...,12(),,,...,( 21"..."...1...1... 100xttsttfttgxsttH njnjnibbbbiccncc jnjnii
from where case 2 is obtained.
3. The edges of In, because of the definition of
n, are of a type:
}),...,1(|),,...,1{( 220
n
nn
n ttstt ,
}1,...,1,,),...(|),,...{( 111 nittttstt ll
n
nn
n
l ,
})0,,...(|),0,,...{( 1111
n
nn
n
n ttstt .
On n
0 there are the sets nKKK ,...,, 21 .
On n
l there are the sets nll KKKK ,...,,,..., 110 .
On n
n there are the sets 110 ,...,, nKKK .
For n
0, we have:
)),2,...,2,,...,2(,12,...,112(),,,...,( 11"...""......1... 100xttstftgxsttH njnibbbbbiccncc jnjnjnii
)),2,...,2,,...,2(,12,...,1(),,,...,( 11"...""......1... 100xttstftgxsttH njnibbbbbiccncc jnjnjnii
)),2,...,2,,...,2(,12,...,12(),,,...,( 11"...""...2...1... 110xttstfttgxsttH njnibbbbbiccncc jnjnjnii
),,,...,( 2...1xsttH ncc , for j
in Kstt ),,...,( 2 , ni ,...,1 .
For n
i , we have two cases, either il or il in the general formula of
cccc nH
110 ... . It cannot happen that il , because on the edge n
i there is no set iK .
For il , we have:
)),2,...,2,,...,2(,12,...,12,12,...,12(),,,...,( 11"..."...11...1... 00xttstfttttgxsttH njnibbbbillccncc jnjnii
)),2,...,2,,...,2(,12,...,12,12,...,12(),,,...,( 11"..."...1...1... 00xttstfttttgxsttH njnibbbbillccncc jnjnii
)),2,...,2,,...,2(,12,...,2,12,...,12(),,,...,( 11"..."...21...ˆ...1... 00xttstfttttgxsttH njnibbbbillcccncc jnjniil
),,,...,,...,,ˆ,...,( 211...ˆ... 110xstttttH nillccccc nl
,
for j
inil Kstttt ),,...,,...,ˆ,...,( 11 , nli ,...,ˆ,...,1 .
For il , we have two cases again, i.e. either both lt and 1lt are before s , or
both lt and 1lt are after s . We will show the case when both lt and 1lt are before
s , and the other can be shown similarly:
)),2,...,2,,...,2,2,...,2(,12,...,12(),,,...,( 111"...""...1...1... 110110xttstttfttgxsttH njnllibbbbbicccncccc jnjnjniin
)),2,...,2,,...,2,2,...,2(,12,...,12(),,,...,( 11"...""...1...1... 110110xttstttfttgxsttH njnllibbbbbicccncccc jnjnjniin
)),2,...,2,,...,2,2,...,2(,12,...,12(),,,...,( 121"...""...ˆ...1...1...1100
xttstttfttgxsttH njnllibbbbbbicccnccjnjnjniii
),,,...,,ˆ,...,,...,( 211...ˆ... 110xstttttH nlliccccc nl
,
for j
inli Kstttt ),,...,ˆ,...,,...,( 11 , nli ,...,ˆ,...,1 .
For n
n , we have:
)),0,...,2,,...,2(,12,...,12(),,,...,( 11"...""...1...1... 110110xtstfttgxsttH jnibbbbbicccncccc jnjnjniin
)),2,...,2,,...,2(,12,...,12(),,,...,( 1111
"..."...1...1... 10110xttstfttgxsttH njni
nbbbbicccncccc jnjniin
),,,...,,...,( 11... 110xstttH niccc n
, за j
in Kstt ),,...,( 11 , for each 1,...,0 ni .
By showing case 3, we have shown the coherence condition for cccc nH
110 ... .
We have also shown that the homotopy cccc nH
110 ... is well defined. Let us also
show that cccc nH
110 ... connects cccc nh
110 ... and cccc nh
110 ..." :
For 0s , by the definition of the sets j
iK , it follows that for each
0, ltnljn , so we have:
)),0,...,0,0,2,...2(,12,...,12(),0,,...,( 11"...""...1...1... 100xttfttgxttH jnibbbbbiccncc jnjnjnii
)),2,...2(,12,...,12(),0,,...,( 11...1...1... 00xttfttgxttH jnibbiccncc jnii
,
for j
in Ktt )0,,...,( 1 , and that corresponds to the definition of cccc nh
110 ... .
For 1s , by the definition of the sets j
iK , it follows that
2
1,, mtjnmim , so we have:
)),2,...,2,2
12,...,
2
12(,12,...,12(),1,,...,( 1"...""...1...1... 1100
xttfttgxttH njnbbbbbicccncc jnjnjnii
)),2,...,2,1,...,1(,12,...,12(),1,,...,( 1"...""...1...1... 1100xttfttgxttH njnbbbbbicccncc jnjnjnii
)),2,...,2(,12,...,12(),1,,...,( 1"...""1...1... 1100xttfttgxttH njnbbbicccncc jnjni
,
for j
in Ktt )1,,...,( 1 , and that corresponds to the definition of cccc nh
110 ..." .
To illustrate, we take the case 2:
We have defined ccch10
and ccch10
" in the following way (picture 2):
)),((2
1)),(,12,12(
)),((2
1)),,2(,12(
)),((2
1),,2,2(
),,(
221221
1212121
021121
21
10
110
100
10
Ktttxfttg
Kttttxtftg
Ktttxttfg
xtth
bccc
bbcc
bbbc
ccc
)),((2
1)),(,12,12(
)),((2
1)),,2(,12(
)),((2
1),,2,2(
),,("
2212"21
121212""1
021121"""
21
10
110
100
10
Ktttxfttg
Kttttxtftg
Ktttxttfg
xtth
bccc
bbcc
bbbc
ccc
The homotopy ZXIH ccc
2:10
, which connects ccch10
and ccch10
" , is
defined on IK i , for each 2,1,0i , by:
)),,2,2,,2(,12,...,12(),,,( 2121"..."...1...21 22010xttstfttgxsttH jibbbbiccccc jjii
for j
iKstt ),,( 21 , ij 2,...,0 .
More specifically, cccH10
is defined as follows, for each IK i , 2,1,0i :
- For IK 0 :
2
02121"""
1
02121""
0
02121"
21
),,(),,2,2,(
),,(),,2,,2(
),,(),,,2,2(
),,,(
1000
1100
100
10
Ksttxttsfg
Ksttxtstfg
Ksttxsttfg
xsttH
bbbbc
bbbbc
bbbbc
ccc
- For IK1 :
1
1212""1
0
1212"1
21),,(),,2,(,12(
),,()),,,2(,12(),,,(
1110
110
10 Ksttxtsftg
KsttxstftgxsttH
bbbcc
bbbcc
ccc
- For IK2 :
0
221"2121 ),,()),,(,12,12(),,,(1010
KsttxsfttgxsttH bbcccccc
On picture 4 we see the way I2 divides itself into three prisms IK 0 ,
IK1 , and IK2 .
Picture 4
On pictures 5, 6 and 7 below we can see the corresponding partitionings of
IK i into j
iK , 2,1,0i .
- IK 0 divides into three parts, 0
0K , 1
0K and 2
0K :
1t
2t
1K
1K
2K
2K
0K
0K
1
1
1
s
ccH0
ccH1
10ccH
Picture 5
- IK1 divides into two parts, 0
1K and 1
1K :
Picture 6
- IK2 does not divide and the one part is 0
2K :
2t
1t
0
1K
1
1
1
s
1
1K
2
1
1t 1
1
1
2
1
2
1
2
0K
1
0K
0
0K
2t
s
Picture 7
Now we may define a composition of classes of coherent proximate nets by:
][][][ fgfg .
The composition of classes of the coherent proximate nets defined in such a
way is well defined.
3. The category of strong shape
Theorem 3.1. If : YXf , : ZYg , ),,(: DwWWZh dd , are
proximate coherent nets, then the proximate coherent nets h gf( ) and ( )hg f are
homotopic.
Proof. Suppose )( bff , )( cgg and )( dhh . In order to obtain an
explicit formula for the proximate coherent net h gf( ) we define a decomposition of n into subpolyhedra mlK , for any pair of integers ml, such that nml0 ,
}.4
1,
2
1|),...,,{( 1121, mmllnml tttttttK
By applying the composition formula twice, for mln Ktt ,1 ),...,( we have
))),4,...,4(,14,...,14(,12,...,12(),())(( 1)...(1)...(1...0xttfttgtthxtgfh kmddghmlddhlddd kmmll
1t
2t
1
1
1
s
0
2K
In the same way, to obtain an explicit formula for the proximate coherent net
)( fhg we define a decomposition of n into subpolyhedra
mlQ , for any pair of
integers ml, such that nml0 , }2
1,
4
3|),...,{( 111, mmllnml ttttttQ .
Then, for mlk Qtt ,1 ),...,( we have
))),2,...,2(,24,...,24(,34,...,34(),())(( 1)...(1)...(1...0xttfttgtthxtfhg nmddghmlddhlddd nmmll
We define a partition of In into subpolyhedra mlM , for any pair of
integers ml, such that nml0 ,
}2
1,
4
2|),,...,{( 111, mmllnml t
stt
ststtM .
We define a homotopy WXIH : which connects h gf( ) and ( )hg f .
This map will be given by the function ADfgh: and by the maps
0)(: ddfgh
n
d WXIHn
defined for mlMst ,),( in the following way:
))),1
4,...,
1
4(,14,...,14(,
2
24,...,
2
24(
),,(
1)...(1)...(
1...0
xs
t
s
tfststg
s
st
s
sth
xstH
nmddghmlddh
ldd
d
nmmll
We mention that })0,,...,(|)0,,...,{(0 ,11, mlnnml MttttK and then, for
mln Ktt ,1 ),...,( , it is easily checked that ),())((),0,( xtgfhxtHj
dd .
Also, })1,,...,(|)1,,...,{(1 ,11, mlnnml MttttQ and for mln Qtt ,1 ),...,( ,
),())((),1,( xtfhgxtH dd .
To the end of the proof we will check the well definition and the coherence
conditions of the map j
dH . To check that the definition is well, suppose that
mlmln MMstt ,1,1 ),,...,( , in which case j
dH is defined in two ways. Then it has to
be that 4
2 stl
. If we compute the formula for mlk Msttt ,1 ),,...,( and 4
2 stl
,
then we have:
))),1
4,...,
1
4(,14,...,14(
,2
24,...,
2
24(),,(
1)...(1)...(
11... 10
xs
t
s
tfststg
s
st
s
sthxstH
nmddghmlddh
lddd
nmmlj
l
The same expression is obtained if we compute the formula for
mln Msttt ,11 ),,...,( and 4
2 stl . Similarly, we can check that the definition is
well for 1,,1 ),,...,( mlmln MMstt , and the other cases can be deduced to one of
these two cases.
To check the coherence conditions of the homotopy dH , suppose that
mln Msttt ,11 ),,...,( and 0nt . Then
)),1
4,...,
1
4(),0,
1
4,...,
1
4( 11
)...(11
)...( 1x
s
t
s
tfx
s
t
s
tf nm
ddghnm
ddgh nmnm, and it follows that for
0nt , ),,...,(),,( 11... 10xttHxstH nddd n
.
The case when 01t is treated similarly.
If 1ii tt , and li , then
),,,...,ˆ,...,(
))),1
4,...,
1
4(,14,...,14(
,2
24,...,
2
24,
2
24,...,
2
24(),,(
1...ˆ...
1)...(1)...(
111...
0
0
xstttH
xs
t
s
tfststg
s
st
s
st
s
st
s
sthxstH
niddd
nmddghmlddh
liiddd
ni
nnmi
i
The cases mil and nim are treated similarly.
Theorem 3.2. If proximate coherent nets :, YXff are homotopic, and
coherent maps :, ZYgg are level homotopic, then the coherent maps
:, YXfgfg are level homotopic.
Proof. Let :, YXff be homotopic by a homotopy : YXIF given
by a strictly increasing function f B A: . Then the proximate coherent nets
:, ZXfggf are homotopic by the homotopy : ZXIgF given by a strictly
increasing function fg C A: .
Let :, ZYgg be homotopic by a homotopy : ZYIG given by the
strictly increasing function g C B: . Then the proximate coherent nets
:, ZZfgfg are homotopic by the homotopy :)'1( ZXIfG given by
strictly increasing function fg C A: .
It follows that the proximate coherent nets :, ZXfgfg are homotopic.
Theorem 3.3. The proximate coherent nets f and 1Xf are homotopic ;
f and 1 fY are homotopic.
Proof. We will prove that f and 1 fY are homotopic, and the other statement
is treated in the similar way.
First we define a partition of In into subpolyhedra lL , nl ,...,1,0 , by
}2
1
2|),,...,{( 11 llnl t
ststtL .
We define a homotopy YXIF : . This map will be given by the
function ABf : and by the maps 0)(: bbf
n
b YXIFn
defined for lLst ),(
by ),1
2,...,
1
2(),,( ... x
s
t
s
tfxstF nl
bbb nl.
We mention that })0,,...,(|)0,,...,{(0 11 lnnl LttttK and then, for
ln Kttt ),...,( 1 , we have ),()1(),0,( xtfxtF bYb .
Also, 011 }),...,(|)1,,...,{( Ltttt n
nn and ),(),1,( xtfxtF bb .
Category of strong shape is obtained. The objects are topological spaces, and
the morphisms are the classes of the coherent proximate nets.
For the isomorphic objects in this category we say they have the same strong
shape.
REFERENCES
[1] K. Borsuk, Theory of Shape, Polish Scientific Publishers, Warszawa, 1975
[2] S. Mardešić, Strong Shape and Homology, Springer Verlag, 1-489, Berlin, 2000,
[3] J.Brendan Quigley, An exact sequence from the n -th to the (n-1)-st fundamental
group, Fund. Math. 77 (1973), no. 3, 195-210
[4] N. Shekutkovski, Intrinsic definition of strong shape for compact metric spaces,
Topology Proceedings 38 (2011)
[5] N. Shekutkovski , Shift and coherent shift in inverse systems, Topology Appl.
Vol. 140, Issue 1 (2004), 111-130
[6] N. Shekutkovski, B. Andonovik, Intrinsic definition of strong shape of strong
paracompacta, Proceedings of IV Congress of the mathematicians of Republic of
Macedonia, Struga, October 18-22, 2008 , (2010), 287 - 299
* Faculty of Mathematics and natural Sciences
** Faculty of Technology and Metallurgy
University of St. Cyril and Methodius
Skopje, Macedonia
email : [email protected]