- Email: [email protected]
Realizing cell-like maps in Euclidean space
chard Departmentof
them&es, UnAwsit.vof Georgia,A rheas, Ga. 36~01, U.S.A.
Abstract: Suppose X and Y are locally compact finite dimensional metric spaces and fi X - Y is a map. Then we say that the map f can be realized: in Euclidean space of dimension k if there elrist closed embeddingsg:X + k and z: Y -+ Ek a;‘< a Josed map h: Ek * Ek such that hg = gfand h IEk \g(X) is a homeom of ‘$\~(XJ onto Ek\g(v). Our main result is that closed cell-like maps can be realized. In particui2r we show, for a closed embeddingg:X+ En, how to obtain a realization off for k = n + diimY + 1. This yields an extension theorem for cell-like maps on closed subsets X of Euclidean sp-ice, where we require that the extension be a horneomorphism on the complement of X, Some: ap cations are givzn in shape theoxy, the prim&y of which is that cell-like maps on finite dimensional EJetric compacta ajie shape preserving. As a tool, we use the notion of comphztely regular mappings as applied to upper semicontiriuous decompositions of manifolds. Some of th:: results obtaic-ed here may be of interest in their own right.
AMS Subj. Class.: Primary 54ClO; Secondary 551199,57A60,57A15 cell-like space cell-like map completely regular mapping
cell-like decomposition of Euclidean space shape fundamental equivalence --
76
general, in that we show that: a a map such as Ir nsion theorem elidean space ism on the complement 0fX. n Section 2 7e give some ~~p~li~ations, “fundamental homstopy category” as desc one of our main results is a Vietoris-type for cell-lrke maps This result statk:s that if f: X * Y is a closed celblike map, where x and Y are finite dimensional separable metric spaces, then ces a fundamental equivalence from f -1 (Z) to Z for each compact 2 c Y. Thus, cell-like maps on compact finite dimeMona metric spaces reserve shape (Theorem 11) and, in particular, the cell-like image or pre-image 04-’a cell-like space is a cel.l-like space (Theorem 8). Our proofs depend on the theory of c 2mpletely regular mappin re31as on some recent resullts of Edwards and Kirby [ 8 1, Armentrout 531, and Siebenmann [ 161. Some of our preliminary results in Section 3 are of interest in their own right. For example, we show there that if C is an upper semicontinuoua cell-like decomposition of EP+Q = g # 4, such that each element of G lies in EPX(y) for some aY such that @dedecomposition of EFX (y) induced by G yields sition space for each y E Eq, then EP+q/G is homeomorphic
is equivalent to an and neighborhood
U off(Y) in Z, there
The map f’:X - Y is said to ccall-like if for each y E Y, f-1 01) is celllike. The reader s cauticmed that a cell4 e map need not be a U *that is, if f: X - Y is eelll-“r e, f -i(y)need not have Pro
z i(.wW
= si
)
CM x G 1)
u
{(O,y)l--1
II),
shall refer to this work several times in the remainde:: of this section. The Mowing rather awkward a pearing technical result will be the basis for the proofs of the main results of this section. . Sugpose that X, Y, and Z are metric spaces and that g:X P:%*Z,anldF:Z+ Yarema (1) the following dia am commutes:
(2) g is completely r (3) for each y E Y,
ists an open set U of Y ccqtaini that A?lg’l (i& g-l (u> --MFW1(u) is uniformly codinu F-1 Q) can be arbitrarily close(4) for each y E Y, Pig-l (y): go1 01)
ly approximated by a homeomorphism.
ty ofg, there exis
ainone of the
mai.n rest.41
jection onto the: seroi~d coordinate. 4-p
Z is a map 0 Spx 89 onto
is a map such hat thi: diagram
mutes. Suppose further that (“y)can be arbitrarily close1 n there is a home~mo~p diagram commutes: f
Suppose Y is a closed subspac.e (tafX which is wtwatcd with respect to G; that is, ifg e G intersects induces an upper semiclontinuous decomposition h a.n induced decom positi co!-Iposition ofXX Y such for each y E Y, then there is a natural map FG : F,_:(g)!24/ ‘fg c e above notation, the following theorem becomes a direct of Theorem 2. core 3. Suppose G is an upper sIemicontinuous decomposition of X 13 such that each element of G lies in SPX {JJ) for some y E I3. Suppose further that P, 4P X (y} * SP X {y)lG’ can be arbitrarily closeIy approximated by a homeomorphism for eaclh y E Eq. Then there is a homleot-florphism $: W X ET + SFX EqjG such that the following diagram corn mutes :
f. Notice first that the hypothesis that PGf can be aribitrarily closely roximated by a homesmorphism is independent of the metric on ’ of G is a closed subset of some SPX
Q
eo
c ~lppose G is a =%4, such that y E FL Suppose further that homeomosp
er sem.icontinuous decomposition t of G lies in SPX (y)for some is a p-manifold for -++ 3% Eq/G sue .
. Corollary 4
W repkced tkn-ough-
ork of tile next e shall not need to appeal to Car Alar;? since, in each case where Armentrr>ut’s or Si esrem might be applied, we calta y homeomor
n-manifolds and
ch that, if i = 2, . . . . piecewise linear Pr-manifold which collapses 1:oan (n!-2)dimensional polydron; see [ 1, Theorem]. f such an X is then regarded as a subs fl +I, by cells; :;ee [ 1, Coroll +I, then X is definable, i ponents of such an X form, along with singletons in En+1\ X, an upper semicontinuous decomposition G ot’P+l. It follows that n+l (see [G]), and that P”: n+l -* En+J/G can be arbitrari closely approximated by a homeomorphism, ith the above facts in hand, we are now ready to state and prove Suppose f: x - Y is a closed cell-like map from the space X onto the r!nite-dimensional space Y, dim Y = W. Suppose g:X -+ El* to a closed subset of Ea. there exists a closed rasp kt ui+m+l such that ::he di .
X !$ EM!$F”pz+m+il
n7;~-inverses,under egenerate elements is
ted by a hsmeomsrw let G denote the upper semicontinuous ,
decomposition
of Enfm”l
each of whose nondegenerate elements is ;SJnon.G’ for some YE I?. (U is upper ~emic~nti~~~~~~
+l and h: En+m+l
Following the proof can be chosen so that p-r@) i in the proof of Theorem 5, $(q,r)
= (q,st_r*(q))
and
noting
with only finitely many nondegener;ite elements. ‘e also note t at the conclusion of Theorem 5 holds dwith plac;eb by !@ GX any k > AI+LYZ + 1. X(v)
TIileorem 5 yields the following extension theorem (compare
[
l&11
] ):
. Su~q~se X is a closed subset of I? and f:X + EI is a closed hat ii’,~ e: f(X), then f -1(p) is cell4ke. Then there exists an ch that j'extends to a closed map f*:Ek ++ Ek such that homeomorphism onto Ek\ f(Xj4If dim f(X) = m and j is an integer suc,h that &X) embeds as a closed subset of E.( then k can be taken any integer ilot less than max {n+m+j+ 1, r+i). . Theorem 5 yields an embedding g: f(X) -+Efl+m+l and a closed map - 3En+malsuch that h I E n+m+r\ f(X) is a homeomorphism and the following diagram commutes: x inci + ,&!$
2 IWX (pz=t,vz+1, r). Then, by [ 111) there exists a ho, neomorphism q+j such that the following 3agxam commutes:
commutesan the map f* =
in some information on embeddin stemming from the exi ows immediately fronn
7. Suppo!;e
and Y are locall e metric spx andfLU++ Yis there exists an iilteger k and embeddings k\E”(Y) are ho tobeZdimX*Cm Y+2. y
The following result follows immediately from Theorem 11 and t main result of { 123. However, it is of interest in its own righ*ka:nd its proof follows sickly from the statemen of ‘lheorem 5. pose X and Y are compact finitead~m~~nsi~~na~ met Y is a cell-like map. T en X is cell-like if and only if
he following is immed.ia-ie from and Y are finiteis a closed cxll-li kt.:
basic notion for the theory is that of fund ntal sequence. Suppose hat follows that X and Y are corn fun&mentalseqvsnce from X to Y is a triple at j& ] is a sequence of maps flh:H -+ M having the property that for each neighborhoqd V of Y there exists a neighborhood U of 31;’such tha,t fk 1U and fk+l1U are rite fkILr~fk+lIU in Y. omotopic in Y for almost = {f&X, 1’) are homotopi ences f = cfk,X, each neighborhood Y there e rists a neighbor X such that fkIUe fi IU in Y for almost all k. The fundamental identity sequence i,. is the sequence &,X,X} obtained b:f taking ik to be the identity H -++H for each k. } and g = &, Y, Z] are fundamental sequences, then the = {gdi,X,Z) is also a fundamental sequence. Now, X and furn&unentalZyequivalent, ortc, have the same shape, if th%ereexistc fittitl9w-mn+n~ ~a xzAIIu~cdlIvllcal ,,quences Iu”= uk, X, Y) and g = [gk, Y9X} such e write X zF Y or #l(X) = Sh(Y). The relation that x and zzlI:is not depe-r t 0-3 any particular choice cf embedding in H and hence induces an equivalent elation, also denoted + or Sh, on metric comacta wh.ich agrees wi homotopy equivalence (in the usual sense) when ; that is, if X, YE ANR, then Sh(X) = Sh(Y) if and on!y if homotopy e ’ alent. The notion of shape has already entered into the study of like spaces via [ 12, Theorem] 1,which states that the finite-dimensional metric continuum -41is czll-like if and ape of a singleton. uppose X and Y are fini dimensional compact - Y is a cell-like ma hen W(X) = Sh(Y).
I$ c V. Let U’ be a neighborhood of Y in _7 such that n( U’) c Int Li. Since g is a fundamental sequence, there e:: sts a neighbcl&osA W of x in H and an integer k, such that i F> k,, ghengk] = gkt_l I W in V@. Now, let iO be an integer such that KioC IV, and let U be a neig.%orhood of Y in H such that n(U) C Int LiO* We now show that if k 2 max (k+ i, ie), then h, 1U 2 &+I 1 U in V, thus showing, since V was compleir3ly arbitrary, that h is a fundamental sequence from Y to X. To see this, let k 2 max (k,, i, io) and 1st G:ktfXb U’beahomotopyfromGO =gkIWtoG1=gk+lI and TV [O,l]. Then (since T(U) C Illt LiO), (by (*), sina;: k 2 ill), (since
i*C 1Iv),
(since G, :
+
(since 7r(U’) C
I/‘),
re:, h, 1U 25h
Hence, HO
d
is generat~~d by the . S>imila: rl y , S:h(7’“). From Theorem 11 we obtain the following “Vietoris-type” theorem. It is roughly the analog of the result t,hat iFf:X -w Y is a cell-like map of locally compact finite-dimensional AWL’s and U C Y is open, then f1f-1{l&f'" (U)* U is a homf.>topy equivalence [ 13,141. and (iii) use some of the results and notions of [ 4, $j 1 1- 151. In (ii), .& denotes tech or Vietoris homology, where G is an Abelian group. In (iii), n, denotes the .tfo fundamental grip of a pair as defined in 114, 5 141; for the proof of (iii) we should perlaps aote that for x0 E X, we can choose the homeomor hisms fi in the proof of Theorem 11 so that ‘j&x($ = P(;;o)* 2. Suppose X and Y are separable finite-dimension spaces, and f:X - Y is a closed cell-like map. If2 is a comya of Y, let W = f “l(Z) and ,g = f1W:cd, 4-b Z. a fundamental e4 ;valenc: ) + fin (Z :G) is an isolm;) = f(wo), then g induces a pointed fundame n] from (W, wO) to (2, zO) and hence an isorrorphfor rz = 1,2 ,....
Alford, uences for compact 0-dimea:donal dec~m~os~tio of En, Bull. Acad. Polon. Sci. S6r.. ys. 17 (1969) 20 -212. Anderson, R.D,, PO ilbert cube, to appear. mentrout, S., Cellular decompositions of 3-manifolds that yield 3-manifolds, Math&Sot. 75 (1969) 453-456. Borsuk, K., Concerning homotopy properties of compacta, Fund. Math, 62 (1968) 223iuk, K., On tie concept of shape for met&able spaces, Bull. Rcad. Polon. Sci. Skx. ath. Astron. Phys. 18 (1970) 127-132. Brown, M., A proof of the generalized Schoenflies theorem, Bull. Amer. Math. SOG.66 (1960) 74-76. Dyer, E, and M. -E. Hamstrom, Completely regular mappings, Fund. Math. 45 (l958j lO3118. Edwards, R.D. and R.C. &by, Deformation< of spaces of embeddings, Ann. Math. (2) 93 (1971) 63-88. Finney, R.L., Uniform limits ol’ compact eeli-tike maps, Notices Amer. Math. Sot. 15 (1968) 942. .H., Extension of homeomorphisms oji Euclidean and Hillbert paraheiotopes, Duke 8 (1941) 452-456. V.L., Jr., Some topological properties uf convex sets, Trans. Amer. 1 Lzeher, RX., Cell-like spaces, Proc. Amer. kT Sot. 20 (1969) 598-60:!. th. 30 Q969) 7i7-7?1. jcher, R.C., Ceil-like mappings, I, Pacific J. ;%fihan, DR., Jr., W properties and relatt;i epics, Mimeographed Idotes,Florida State K ziversity 1970. Sersntiv, M., Characterization of dimension e.bfmetric spaces by continuous mappings into Ekichdean spaces, Uspehi Mat. Nauk 12 (1957) 245-247 (in Ru:sian). Sic?benmann. L.C., Approximating ce by homeomorphisms: 10 appear. Whyburn, G.T., Analytic topology ( . SW, Providence, R.1 ,1963), burn, G.T., Decompositi ‘I‘opology of 3-manifolds ar. :I relate . Fort, Jr. (Prentice-Hall,
them&es, UnAwsit.vof Georgia,A rheas, Ga. 36~01, U.S.A.
Abstract: Suppose X and Y are locally compact finite dimensional metric spaces and fi X - Y is a map. Then we say that the map f can be realized: in Euclidean space of dimension k if there elrist closed embeddingsg:X + k and z: Y -+ Ek a;‘< a Josed map h: Ek * Ek such that hg = gfand h IEk \g(X) is a homeom of ‘$\~(XJ onto Ek\g(v). Our main result is that closed cell-like maps can be realized. In particui2r we show, for a closed embeddingg:X+ En, how to obtain a realization off for k = n + diimY + 1. This yields an extension theorem for cell-like maps on closed subsets X of Euclidean sp-ice, where we require that the extension be a horneomorphism on the complement of X, Some: ap cations are givzn in shape theoxy, the prim&y of which is that cell-like maps on finite dimensional EJetric compacta ajie shape preserving. As a tool, we use the notion of comphztely regular mappings as applied to upper semicontiriuous decompositions of manifolds. Some of th:: results obtaic-ed here may be of interest in their own right.
AMS Subj. Class.: Primary 54ClO; Secondary 551199,57A60,57A15 cell-like space cell-like map completely regular mapping
cell-like decomposition of Euclidean space shape fundamental equivalence --
76
general, in that we show that: a a map such as Ir nsion theorem elidean space ism on the complement 0fX. n Section 2 7e give some ~~p~li~ations, “fundamental homstopy category” as desc one of our main results is a Vietoris-type for cell-lrke maps This result statk:s that if f: X * Y is a closed celblike map, where x and Y are finite dimensional separable metric spaces, then ces a fundamental equivalence from f -1 (Z) to Z for each compact 2 c Y. Thus, cell-like maps on compact finite dimeMona metric spaces reserve shape (Theorem 11) and, in particular, the cell-like image or pre-image 04-’a cell-like space is a cel.l-like space (Theorem 8). Our proofs depend on the theory of c 2mpletely regular mappin re31as on some recent resullts of Edwards and Kirby [ 8 1, Armentrout 531, and Siebenmann [ 161. Some of our preliminary results in Section 3 are of interest in their own right. For example, we show there that if C is an upper semicontinuoua cell-like decomposition of EP+Q = g # 4, such that each element of G lies in EPX(y) for some aY such that @dedecomposition of EFX (y) induced by G yields sition space for each y E Eq, then EP+q/G is homeomorphic
is equivalent to an and neighborhood
U off(Y) in Z, there
The map f’:X - Y is said to ccall-like if for each y E Y, f-1 01) is celllike. The reader s cauticmed that a cell4 e map need not be a U *that is, if f: X - Y is eelll-“r e, f -i(y)need not have Pro
z i(.wW
= si
)
CM x G 1)
u
{(O,y)l--1
II),
shall refer to this work several times in the remainde:: of this section. The Mowing rather awkward a pearing technical result will be the basis for the proofs of the main results of this section. . Sugpose that X, Y, and Z are metric spaces and that g:X P:%*Z,anldF:Z+ Yarema (1) the following dia am commutes:
(2) g is completely r (3) for each y E Y,
ists an open set U of Y ccqtaini that A?lg’l (i& g-l (u> --MFW1(u) is uniformly codinu F-1 Q) can be arbitrarily close(4) for each y E Y, Pig-l (y): go1 01)
ly approximated by a homeomorphism.
ty ofg, there exis
ainone of the
mai.n rest.41
jection onto the: seroi~d coordinate. 4-p
Z is a map 0 Spx 89 onto
is a map such hat thi: diagram
mutes. Suppose further that (“y)can be arbitrarily close1 n there is a home~mo~p diagram commutes: f
Suppose Y is a closed subspac.e (tafX which is wtwatcd with respect to G; that is, ifg e G intersects induces an upper semiclontinuous decomposition h a.n induced decom positi co!-Iposition ofXX Y such for each y E Y, then there is a natural map FG : F,_:(g)!24/ ‘fg c e above notation, the following theorem becomes a direct of Theorem 2. core 3. Suppose G is an upper sIemicontinuous decomposition of X 13 such that each element of G lies in SPX {JJ) for some y E I3. Suppose further that P, 4P X (y} * SP X {y)lG’ can be arbitrarily closeIy approximated by a homeomorphism for eaclh y E Eq. Then there is a homleot-florphism $: W X ET + SFX EqjG such that the following diagram corn mutes :
f. Notice first that the hypothesis that PGf can be aribitrarily closely roximated by a homesmorphism is independent of the metric on ’ of G is a closed subset of some SPX
Q
eo
c ~lppose G is a =%4, such that y E FL Suppose further that homeomosp
er sem.icontinuous decomposition t of G lies in SPX (y)for some is a p-manifold for -++ 3% Eq/G sue .
. Corollary 4
W repkced tkn-ough-
ork of tile next e shall not need to appeal to Car Alar;? since, in each case where Armentrr>ut’s or Si esrem might be applied, we calta y homeomor
n-manifolds and
ch that, if i = 2, . . . . piecewise linear Pr-manifold which collapses 1:oan (n!-2)dimensional polydron; see [ 1, Theorem]. f such an X is then regarded as a subs fl +I, by cells; :;ee [ 1, Coroll +I, then X is definable, i ponents of such an X form, along with singletons in En+1\ X, an upper semicontinuous decomposition G ot’P+l. It follows that n+l (see [G]), and that P”: n+l -* En+J/G can be arbitrari closely approximated by a homeomorphism, ith the above facts in hand, we are now ready to state and prove Suppose f: x - Y is a closed cell-like map from the space X onto the r!nite-dimensional space Y, dim Y = W. Suppose g:X -+ El* to a closed subset of Ea. there exists a closed rasp kt ui+m+l such that ::he di .
X !$ EM!$F”pz+m+il
n7;~-inverses,under egenerate elements is
ted by a hsmeomsrw let G denote the upper semicontinuous ,
decomposition
of Enfm”l
each of whose nondegenerate elements is ;SJnon.G’ for some YE I?. (U is upper ~emic~nti~~~~~~
+l and h: En+m+l
Following the proof can be chosen so that p-r@) i in the proof of Theorem 5, $(q,r)
= (q,st_r*(q))
and
noting
with only finitely many nondegener;ite elements. ‘e also note t at the conclusion of Theorem 5 holds dwith plac;eb by !@ GX any k > AI+LYZ + 1. X(v)
TIileorem 5 yields the following extension theorem (compare
[
l&11
] ):
. Su~q~se X is a closed subset of I? and f:X + EI is a closed hat ii’,~ e: f(X), then f -1(p) is cell4ke. Then there exists an ch that j'extends to a closed map f*:Ek ++ Ek such that homeomorphism onto Ek\ f(Xj4If dim f(X) = m and j is an integer suc,h that &X) embeds as a closed subset of E.( then k can be taken any integer ilot less than max {n+m+j+ 1, r+i). . Theorem 5 yields an embedding g: f(X) -+Efl+m+l and a closed map - 3En+malsuch that h I E n+m+r\ f(X) is a homeomorphism and the following diagram commutes: x inci + ,&!$
2 IWX (pz=t,vz+1, r). Then, by [ 111) there exists a ho, neomorphism q+j such that the following 3agxam commutes:
commutesan the map f* =
in some information on embeddin stemming from the exi ows immediately fronn
7. Suppo!;e
and Y are locall e metric spx andfLU++ Yis there exists an iilteger k and embeddings k\E”(Y) are ho tobeZdimX*Cm Y+2. y
The following result follows immediately from Theorem 11 and t main result of { 123. However, it is of interest in its own righ*ka:nd its proof follows sickly from the statemen of ‘lheorem 5. pose X and Y are compact finitead~m~~nsi~~na~ met Y is a cell-like map. T en X is cell-like if and only if
he following is immed.ia-ie from and Y are finiteis a closed cxll-li kt.:
basic notion for the theory is that of fund ntal sequence. Suppose hat follows that X and Y are corn fun&mentalseqvsnce from X to Y is a triple at j& ] is a sequence of maps flh:H -+ M having the property that for each neighborhoqd V of Y there exists a neighborhood U of 31;’such tha,t fk 1U and fk+l1U are rite fkILr~fk+lIU in Y. omotopic in Y for almost = {f&X, 1’) are homotopi ences f = cfk,X, each neighborhood Y there e rists a neighbor X such that fkIUe fi IU in Y for almost all k. The fundamental identity sequence i,. is the sequence &,X,X} obtained b:f taking ik to be the identity H -++H for each k. } and g = &, Y, Z] are fundamental sequences, then the = {gdi,X,Z) is also a fundamental sequence. Now, X and furn&unentalZyequivalent, ortc, have the same shape, if th%ereexistc fittitl9w-mn+n~ ~a xzAIIu~cdlIvllcal ,,quences Iu”= uk, X, Y) and g = [gk, Y9X} such e write X zF Y or #l(X) = Sh(Y). The relation that x and zzlI:is not depe-r t 0-3 any particular choice cf embedding in H and hence induces an equivalent elation, also denoted + or Sh, on metric comacta wh.ich agrees wi homotopy equivalence (in the usual sense) when ; that is, if X, YE ANR, then Sh(X) = Sh(Y) if and on!y if homotopy e ’ alent. The notion of shape has already entered into the study of like spaces via [ 12, Theorem] 1,which states that the finite-dimensional metric continuum -41is czll-like if and ape of a singleton. uppose X and Y are fini dimensional compact - Y is a cell-like ma hen W(X) = Sh(Y).
I$ c V. Let U’ be a neighborhood of Y in _7 such that n( U’) c Int Li. Since g is a fundamental sequence, there e:: sts a neighbcl&osA W of x in H and an integer k, such that i F> k,, ghengk] = gkt_l I W in V@. Now, let iO be an integer such that KioC IV, and let U be a neig.%orhood of Y in H such that n(U) C Int LiO* We now show that if k 2 max (k+ i, ie), then h, 1U 2 &+I 1 U in V, thus showing, since V was compleir3ly arbitrary, that h is a fundamental sequence from Y to X. To see this, let k 2 max (k,, i, io) and 1st G:ktfXb U’beahomotopyfromGO =gkIWtoG1=gk+lI and TV [O,l]. Then (since T(U) C Illt LiO), (by (*), sina;: k 2 ill), (since
i*C 1Iv),
(since G, :
+
(since 7r(U’) C
I/‘),
re:, h, 1U 25h
Hence, HO
d
is generat~~d by the . S>imila: rl y , S:h(7’“). From Theorem 11 we obtain the following “Vietoris-type” theorem. It is roughly the analog of the result t,hat iFf:X -w Y is a cell-like map of locally compact finite-dimensional AWL’s and U C Y is open, then f1f-1{l&f'" (U)* U is a homf.>topy equivalence [ 13,141. and (iii) use some of the results and notions of [ 4, $j 1 1- 151. In (ii), .& denotes tech or Vietoris homology, where G is an Abelian group. In (iii), n, denotes the .tfo fundamental grip of a pair as defined in 114, 5 141; for the proof of (iii) we should perlaps aote that for x0 E X, we can choose the homeomor hisms fi in the proof of Theorem 11 so that ‘j&x($ = P(;;o)* 2. Suppose X and Y are separable finite-dimension spaces, and f:X - Y is a closed cell-like map. If2 is a comya of Y, let W = f “l(Z) and ,g = f1W:cd, 4-b Z. a fundamental e4 ;valenc: ) + fin (Z :G) is an isolm;) = f(wo), then g induces a pointed fundame n] from (W, wO) to (2, zO) and hence an isorrorphfor rz = 1,2 ,....
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