a-Homotopy and a-Homotopy Type Spaces

Text Only Version

?-Homotopy and ?-Homotopy Type Spaces

Department of Mathematics,

Rani Channamma University, Belagavi-591 156, Karnataka, India.

T Venkatesh

Department of Mathematics,

Rani Channamma University, Belagavi-591 156, Karnataka, India.

Abstract- In this paper, we introduce the term of -homotopy of

-continuous maps,-homotopy equivalence classes and their consequences, same -homotopy type, -contractible spaces, and few properties. Also we present induced map and their properties, and consequences respect to some notions.

Key Words: -homotopy, same -homotopy type, -contractible space.

AMS Classication: 54C08, 14F35, 55Q05, 55P15, 55U35.

1. INTRODUCTION

Algebraic methods were introduced in topology by Poincare around 1895. Chronologically, fundamental group of topological spaces and homology groups were some of important notions that enable us to compute for the topological spaces (Analysis Situs, 1895) [7, 9, 12]. As a result a separate branch took birth known as algebraic topology. Later, we see in the work of Poul Heegard, Barratt etc, in the extensive reading of homotopy and their equivalence among the spaces M and N. Well, there algebraic methods enhanced the scope of the classification themes among topological spaces.

Next to these algebraic notions, we have certain aspect of associations not necessarily a algebraic structure with topological space, namely the notion homotopy equivalence classes over space of all continuous function from a space to space. Deformations being standard idea in topology which in a way converts one space into anther space without losing its qualitative property. Homotopy equivalence classis set [M, N] was first systemically studied by M.G. Barratt in 1955 [9]. The issue of characterization of homeomorphic topological spaces has partially solved by help of homotopy. The problems of classification of topological spaces according to their homotopy properties are also take important role in classification theory. But homotopy equivalence of spaces not necessarily topological equivalence. Homotopy equivalence is a weaker relation than topological equivalence, hence its class is bigger. Therefore homotopy equivalence has prominent role than homeomorphism. Deformation is a standard idea in topology, as morphisms can do so. The deformation is a notion that converts one space into other by the tool of homotopy. Hence homotopy theory takes powerful tools for this purpose [1,4,8,9]. In geometry, not only homotopy as well homology and cohomology are frequently used algebraic association.

Njastad [3,11] introduced -open sets in a topological space and studied some properties. Further – continuous map, and respective open map and closed map in topological space is studied by A.S.Mashhour, I.A.Hasanein

introduce such class of -homotopy of -continuous maps, same -homotopy type, -contractible space, and post and pre induced map of a -continuous map. These have many rich consequences concern to contractible space, induced map, usual Homotopy and fundamental group. Here we place example for few respective results.

2. PRELIMINARIES

Throughout this paper L, M, N, R and S represent the topological spaces on which no separation axioms are assumed unless otherwise mentioned. Map here mean function and for a subset F of topological space M, the M\F denotes the complement of F in M. We recall the following definitions.

Definition 2.1 [3, 11] A subset F of a space M is said to be – open set, if F Int(Cl(Int(F))). Respectively called -closed set, if M F is -open set in M. We denote the set of all – open sets in M by O(M).

Definition 2.2 [2] A map h: M N is said to be – continuous map, if p(F) is -closed set of M, for every closed set F of N.

Definition 2.3 A map h: M N is said to be -irresolute map, if p(F) is -closed set of M, for every -closed set Fof N.

Definition 2.4 A bijection h: M N is called – homeomorphism, if both h and p are -continuous.

Definition 2.5 (Consider) A topological space M is called – space, if every -closed set is closed set.

Theorem 2.6 Every -irresolute maps are -continuous.

3. -HOMOTOPY

In this section we introduce the notions of – homotopy of -continuous maps, -relative homotopy, – contractible space, -homotopy type and post and pre induced map of a -continuous map also discuss few properties respectively.

Denition 3.1 Let M and N be two topological spaces and g, h: M N be two -continuous maps. Then a -homotopy is a map H: M Ã— I N (Here, I = [0, 1]) such that, for all t I the restrictions of H, Ht: M N by x Ht(x) = H(x, t) is

-continuous, satisfying H0(x) = H(x, 0) = g(x) and H1(x) = H(x, 1) = h(x) for all x M. If such H is exist then H is called -homotopy between them, and g is called –

homotopic to h. We denote it by g h.

And If g is -homotophic to a constant map ( i. e. g cy , where cy: M N, cy(x) = y for some y N) then g is called -nullhomotopic.

Example 3.2 Let g, h: (, ) (, ) be g(x) = 5x ,

[2]. Semi-homotopy and semi-fundamental groups studied by u u

Ayhan Erciyes, Ali Aytek in and Tuncar Sahan [4]. We

h(x) = x + 2 and ube usual topology on , then obviously map g is become -homotopic to h, quite simple that one can

check H(x, t) = (1 t) 5x + t (x + 2) is a -homotopy, because we know every continuous maps are -continuous.

Theorem 3.3 Every homotopy is -homotopy (Every homotopy between continuous maps is -homotopy between same -continuous maps.).

Proof: It fallowed from the fact, every continuous map is – continuous map.

Theorem 3.4 [1,6,8] The relation -homotopic is an equivalence relation on the set -C(M, N) of all -continuous maps from topological space M to N.

Proof: Let M and N be two topological spaces then, Reexivity: If g -C(M, N) i.e. g: M N is -continuous map. Dene H: M Ã— I N by H(x, t) = g(x) for all x M and all t I, then H become -homotopy between g and g. Hence g g.

Symmetry: Suppose g, h -C(M, N) and g h implies

there is a -homotopy map H: M Ã— I N such that, for all t I the restrictions of H, Ht: M N by x Ht(x) = H(x, t) is -continuous, satisfying H(x, 0) = g(x) and H(x, 1) = h(x) for all x M . Dene F: M Ã— I N , by F(x, t) = H(x, (1 t)), obviously for all t I the restrictions of F ,

Ft: M N by x Ft(x) = H(x, (1 t)) = H(x, t0) = Ht

Theorem 3.8 If g C(M, N) then [g] [g] where [g] is homotopy equivalence class of g . But C(M, N) and -C(M, N) are not nessasarly comparable.

Proof: Since every continuous map is -continuous and

converse need not hold.

Lemma 3.9 [9] If g, h: M N are -continuous maps,

g h and k: N R is continuous then kog koh.

Proof: The g h , of g, h: M N -continuous maps guarantee that there is a -homotopy H: M Ã— I N such that, for all t I the restrictions of H, Ht: M N by x Ht(x) = H(x, t) is -continuous, satisfying H(x, 0) = g(x) and H(x, 1) = h(x) for all x M . Now define the map F = koH: M Ã— I R clearly, for all t I the restrictions of F , Ft: M N by x Ft(x) = koH(x, t) = koHt(x) is – continuous because H is -continuous and k continuous. Also satisfying F(x, 0) = kog(x) and F(x, 1) = koh(x) for all x

M. Hence the result.

Theorem 3.10 If N is -space, g, h: M N are -continuous maps, g h and k: N R is -continuous then kog koh.

Proof: Similar to lema 3.9.

for some

, hence it is

0

-continuous, satisfyin

Theorem 3.11 If N is -space, g, h: M N are -irresolute

t0 I

g maps, g

h and k: N R is -continuous then

F(x, 0) = h(x) and F(x, 1) = g(x) and become -homotopy

between h and g, implies h g.

Transitivity: Suppose g, h, k -C(M, N) with g h and h k implies there are – homotopies, H: M Ã— I N such that, for all t I the restrictions of H , Ht: M N by x Ht(x) = H(x, t) is -continuous, satisfying H(x, 0) = g(x) and H(x, 1) = h(x) for all x M . And F: M Ã— I N such

that, for all t I the restrictions of F , Ft: M N by x Ft(x) = F(x, t) is -continuous, satisfying with F(x, 0) = h(x) and F(x, 1) = k(x) for all x M. Dene map G: M Ã— I N by,

H(x, 2t) if t [0,1/2]

G(x, t) = { . Obvious G become –

F(x, 2t 1) if t [1/2,1]

homotopy between g and k , because Gt(x) = G(x, t) =

kog koh.

Proof: Refer lemma 3.9.

Lemma 3.12 [9] If N is -space g, h: N R are – continuous maps, and g h and f: M N is -continuous then gof hof.

Proof: Hypothesis is g, h: N R are -continuous maps, and g h. This implies there is -homotopy H: N Ã— I R such that, for all t I the restrictions of H , Ht: N R by y Ht(y) = H(y, t) is -continuous satisfying H0(y) =

H(y, 0) = g(y) and H1(y) = H(y, 1) = h(y) for all y N . Define F: M Ã— I R by F(x, t) = H(f(x), t) , this is well

define map and F0(x) = H(f(x), 0) = gof(x) and F1(x) = H(f(x), 1) = hof(x) for all x M . Also for all t I the

restrictions of F is, F : M R by x F (x) = H(f(x), t) =

Ht or Ft for some t0 I, hence it is -continuous, satisfying t t

0 0 Ht(f(x)) = Htof(x) which is -continuous because of N is

G(x, 0) = g(x) and F(x, 1) = k(x) and become -homotopy,

implies g k. Therefore is an equivalence relation. Equivalence of -homotopy on -C(M, N) gives following notions.

Denition 3.5 Let M and N be two topological spaces and g: M N be -continuous map. Then the set of all – continuous maps from M to N which are -homotopic to g is called homotopy equivalence class of g. It is denoted by [g] or -Eg.

1. [g] = {h -C(M, N) g h}.

Denition 3.6 Let M and N be two topological spaces, then the set of all -homotopy equivalence classes over -C(M, N)

is called -homotopy equivalence classes over -C(M, N).

This is denoted by -C(M, N) or [M, N] . That is

-space and f: M N is -continuous.

Theorem 3.13 If N is -space, g, h: N R are -irresolute maps, g h and f: M N is -continuous then gof hof. Proof: Since every -irresolute map is -continuous map.

Lemma 3.14 [9] If N is -space, f, g: M N and h, k: N R are -continuous maps and f g, h k then hof kog. Proof: Hypothesis is f, g: M N and h, k: N R are – continuous maps and f g, h k. This implies there is – homotopy H(x, t) between f and g, such that, for all t I the restrictions of H , Ht: M N by x Ht(x) = H(x, t) is – continuous maps satisfying H0(x) = H(x, 0) = f(x) and H1(x) = H(x, 1) = g(x) for all x M . Also there is – homotopy G(y, t) between h and k, such that, for all t I the

restrictions of G , G : M N by y G (y) = G(y, t) is –

t t

-C(M, N) = {[g]

g -C(M, N)}.

continuous map satisfying G0(y) = G(y, 0) = h(y) and

Theorem 3.7 If topological space M is -space then -C(M, N) = C(M, N) (Here C(M, N denotes the set of all continuous maps from M to N).

Proof: Open sets and -open sets are same in -space.

G1(y) = G(y, 1) = k(y) for all y N. Define F: M Ã— I R as F(x, t) = G(H(x, t), t) , which become map and F(x, 0) = G(H(x, 0), 0) = hof(x) , F(x, 1) = G(H(x, 1), 1) = kog(x) .

Since Ht and Gt are -continuous map and N is -space gives GtoHt is -continuous map and GtoHt(x) =

Gt(Ht(x)) = G(H(x, t), t) = Ft(x) = F(x, t). Hence for all t I the restrictions of F, Ft: M R by x Ft(x) = F(x, t) is – continuous map. Therefore F is -homotopy between hof and kog.

Corollary 3.15 [9] If N is -space p: N R is -continuous and for all f -C(M, N) then there exist map p: [M, N] [M, R] by p([f]) = [pof], (call pre induced map).

Proof: Here N is -space p: N R is -continuous. Defined p: [M, N] [M, R] by p([f]) = [pof] , is clearly well defined because, [M, N] implies = [f] for some f: M N -continuous. This gives pof: M R is – continuous, implies[pof] = p([f]) [M, R].

Also suppose for [M, N] such that = [f] = [g] , implies f g by theorem 3.10 pof pog , so [pof] = [pog] and p([f]) = p([g]). Hence p is well defined.

Theorem 3.16 If N, R are -spaces, p: N R, q: R S are -continuous maps and for all [M, N] then (qop)() = qop().

Proof: N, R are -space and p: N R , q: R S are – continuous maps implies qop: N S is -continuous. Therefore each map induce p: [M, N] [M, R] , q: [M, R] [M, S] and (qop): [M, N] [M, S] , hence qop possible also qop and (qop) have same domain and codamain.

Also [M, N] implies = [f] for some f: M N -continuous. Consider,

(qop)() = (qop)([f])

= [(qop)of]

= [qo(pof)]

= q([pof])

= qop([f])

= qop()

Theorem 3.17 If N is -space, IdN: N N defined as IdN(x) = x and for all f -C(M, N) then (IdN)([f]) = Id[M,N]([f]), [f] [M, N].

Proof: Here N is -space, IdN: N N defined as IdN(x) = x

is -continuous. This implies pre induced map become,

(IdN): [M, N] [M, N] by (IdN)([f]) = [IdNof] =

[f] = Id[M,N]([f]), [f] [M, N]. Hence the proof. Corollary 3.18 [9] If N is -space q: M N is -continuous and for all f -C(N, R) then there exist map q#: [N, R] [M, R] by q#([f]) = [foq] (call post induced map).

Proof: Here Nis -space, q: M N is -continuous. Defined q#: [N, R] [M, R] by q#([f]) = [foq] , is clearly well defined because, [N, R] implies = [f] for some f: N R -continuous. This gives foq: M R is – continuous, implies [foq] = q#([f]) [M, R].

Also suppose for [N, R] such that = [f] = [g], implies f g by lemma 3.12 foq goq, so [foq] = [goq] and q#([f]) = q#([g]). Hence q# is well defined.

Theorem 3.19 If M, N are -space, p: L M, q: M N are -continuous maps and for all [N, R] then (qop)#() = p#oq#().

Proof: M, N are -space and p: L M , q: M N are – continuous maps implies qop: L N is -continuous. Therefore each map induce p#: [M, R] [L, R] , q#: [N, R] [M, R] and (qop)#: [N, R] [L, R] , hence

p#oq# possible also p#oq# and (qop)# have same domain and codamain.

Also [N, R] implies = [f] for some f: N R -continuous. Consider,

(qop)#() = (qop)#([f])

= [fo(qop)]

= [(foq)op]

= p#([foq])

= p#oq#([f])

= p#oq#()

Theorem 3.20 If M is -space, IdM: M M defined as IdM(x) = x and for all f -C(M, N) then (IdM)#([f]) = Id[M,N]([f]), [f] [M, N].

Proof: Here M is -space, IdM: M M defined as IdM(x) = x is -continuous. This implies post induced map,

(IdM)#: [M, N] [M, N] become (IdM)#([f]) =

[(foIdM)] = [f] = Id[M,N]([f]), [f] [M, N]. Hence the proof.

Denition 3.21 Let M and N be two topological spaces A M and g, h: M N be two -continuous maps. Then a – relative homotopy respect to A is a map H: M Ã— I N such

that, for all t I the restrictions of H , Ht: M N by x Ht(x) = H(x, t) is -continuous, satisfying H0(x) = H(x, 0) = g(x) , H1(x) = H(x, 1) = h(x) for all x M and H(a, t) = g(a) = h(a), a A, t I. If such H is exist then

H is called -relative homotopy between them, and g is called -relative homotopic to h respect to A . We denote it by g (A) h.

Example 3.22 Let g, h: be g(x) = x and

1. if x 0

h(x) = {x if x [0,1] then g (A) h, where A = [0,1].

2. if x 1

Theorem 3.23 Every relative homotopy respect to A is –

relative homotopy respect to A. Proof: Refer theorem 3.3.

Theorem 3.24 -relative homotopy respect to A is an equivalence relation on set of all -continuous maps from M to N.

Proof: Refer theorem 3.4.

Theorem 3.25 [9] If N is -space, f, g: M N are -relative homotopy respect to A M, and h, k: N R are -relative homotopy respect to B N and f(A) B then hof (A) kog.

Proof: Hypothesis f, g: M N and, h, k: N R are – continuous maps and f (A) g, h (B) k. This implies there is -relative homotopy H(x, t) respect to A, between f and g, such that, for all t I the restrictions of H, Ht: M N by

x Ht(x) = H(x, t) is -continuous maps satisfying H0(x) = H(x, 0) = f(x), H1(x) = H(x, 1) = g(x) for all x M and H(a, t) = f(a) = g(a), a A, t I. Also there is –

relative homotopy G(y, t) respect to B, between h and k, such that, for all t I the restrictions of G , Gt: M N by y

Gt(y) = G(y, t) is -continuous map satisfying G0(y) = G(y, 0) = h(y) , G1(y) = G(y, 1) = k(y) for all y N and

G(b, t) = h(b) = k(b), b B, t I.

Define F: M Ã— I Ras F(x, t) = G(H(x, t), t)become map and F(x, 0) = G(H(x, 0), 0) = hof(x) , F(x, 1) =

G(H(x, 1), 1) = kog(x) and F(a, t) = G(H(a, t), t) =

G(f(a), t) = hof(a) = kof(a) = kog(a), a A, t I. Since Ht and Gt are -continuous map and N is -space gives GtoHt is -continuous map and GtoHt(x) = Gt(Ht(x) =

G(H(x, t), t) = Ft(x) = F(x, t) . Hence for all t I the restrictions of F , Ft: M R by x Ft(x) = F(x, t) is – continuous map. Therefore F is -relative homotopy respect

to A between hof and kog .

Denition 3.26 Let M and N be two topological spaces and g: M N be -continuous map if there is -continuous map h: M N such that hog IdM and goh IdN . Then h is called -homotopy equivalence to g, or -homotopy inverse of g. Here g, h are called -homotopy equivalences between M and N.

Denition 3.27 Let M and N be two topological spaces, if there is a -homotopy equivalences between them, then M and N are called -homotopy equivalent to each other or same -homotopy type.

Example 3.28 The cylinder and circle are same -homotopy type. It will follow by theorem 3.29.

Theorem 3.29 Same homotopy type spaces are also same – homotopy type spaces. Converse need not hold.

Proof: Fallowed by the theorem 3.3.

Note 3.30 Generally composition of two -continuous maps are need not -continuous. Therefore same -homotopy type relation even thou reflexive, symmetric but not transitive, so it will not induce equivalence relation. Therefore our intuition, partition of topological space under same – homotopy type is not possible. Since every -irresolute map are -continuous map and composition of two -irresolute maps are -irresolute, also confining set of all topological spaces are -spaces on which the relation same -homotopy type become equivalence relation as fallow.

Theorem 3.31 [9] The relation same -homotopy type is equivalence relation on set of – topological spaces.

Proof: Reflexive: for every – space M, obvious IdM, IdM is a -homotopy equivalences.

Symmetric: Let M, N and R are -spaces, suppose M and N are same -homotopy type, implies there is g, h a – homotopy equivalences. Obvious h, g gives -homotopy equivalences between N and M.

Transitive: Let M, N and R are -spaces and M, N are same -homotopy type and N, R are same -homotopy type. This implies there exist a -homotopy equivalences f, g between M and N, that is gf IdMand fg IdN via a -homotopy. Also there exist a -homotopy equivalences h, k between N and R , such that kh IdN and hk IdR via a – homotopy. It is true that hf: M R and gk: R M become -homotopy equivalences between M and R. Because since the hypothesis kh IdN and fact f f, and by the lemma

3.14 we have khf IdNf again by lemma 3.14 we can have gkhf gIdNf , this implies gkhf gIdNf IdM by hypothesis. Similarly

ii) If p: N R is -homeomorphism and for any space M

then p: [M, N] [M, R] is bijective.

Proof: i) Here p: N R is -homotopy equivalence implies existence of its -homotopy inverse say q: R N such that poq IdR and qop IdN . For any space M , since

p: [M, N] [M, R] is well defined function, suppose p([f]) = p([g]) for [f], [g] [M, N] , gives [pof] = [pog] which implies pof pog. The hypothesis q: R N is -continuous, R is -space and pre composition theorem

3.10 implies qopof qopog equivalently f g so

[f] = [g], therefore p is injective.

Also for any [h] [M, R] implies h: M R is – continuous map, and hypothesis q: R N is -continuous, R is -space guarantee qoh: M N is -continuous. Therefore [qoh] [M, N] such that p([qoh]) = [poqoh] = [h] , hence surjective.

ii) p: N R is -homeomorhism then p and p1 become – homotopy equivalences.

Theorem 3.33 [9,12] If N, M are -space then,

1. q: M N is a -homotopy equivalence and for any space R

then q#: [N, R] [M, R] is bijective.

2. q: M N is -homeomorhism and for any space R then

q#: [N, R] [M, R] is bijective.

Proof: i) Here q: M N is -homotopy equivalence implies existence of its -homotopy inverse say p: N M such that

poq Id M and qop IdN . For any space M , since q#: [N, R] [M, R] is well defined function, suppose q#([f]) = q#([g]) for [f], [g] [N, R] , gives [foq] = [goq] which implies foq goq. The hypothesis p: N M is -continuous, Mis -space and post composition lemma

3.12 implies foqop goqop equivalently f g so

[f] = [g], therefore q# is injective.

Also for any [h] [M, R] implies h: M R is – continuous map, and hypothesis p: N M is -continuous, M is -space guarantee hop: N R is -continuous. Therefore [hop] [N, R] such that q#([hop]) = [hopoq] = [h] , hence surjective.

ii) q: N R is -homeomorhism then q and q1 become – homotopy equivalences.

Definition 3.34 A topological space M is called – contractible if IdM is -null homotophic.

Theorem 3.35 Every contractible space is -contractible. Proof: Since every homotopy is -homotopy.

Example 3.36 Every star convex space is -contractible. Theorem 3.37 If M is -contractible then it is same – homotopy type to a point space in M.

Proof: Let M be a -contractible space, implies identity map IdM on M is -null homotopic. That is there exist cx0 : M M by cx0 (x) = x0, such that IdM cx0 . Define h: {x0} M by

h(x) = x0, and g: M {x0} by g(x) = x0. Then hog: M M

become hog(x) = x0 = cx , so hog = cx IdM , also

hfgk Id . Therefore hf and gk are -homotopy 0 0

R

equivalen en

and

. Hence same

-homotopy

goh: {x0} {x0} become goh(x) = x0 = Id{x0} , so

ces betwe M R

goh(x)

Id{

} (x) . Therefore {x } and M are same –

type is equivalence relation.

Theorem 3.32 [9, 12] If N, R are -space then,

i) If p: N R is a -homotopy equivalence and for any space

M then p: [M, N] [M, R] is bijective.

x0 0

homotopy type.

Theorem 3.38 Every -contractible space is same – homotopy type of a point space.

Proof: Let M be a -contractible space, and {} be any point space. By theorem 3.37 there exist a point space {x0} in M

such that {x0} and M are same -homotopy type. Since {x0} and {} are homeomorphic space, hence same -homotopy type. Here transivity works by same -homotopy type relation gives M is same -homotopy type of point space {}.

Theorem 3.39 If N is -contractible -spce and for any space M then [M, N] is singleton.

Proof: Consider any g, h -C(M, N) , implies g, h M N are -continuous maps. Hypothesis is N is -contractible space implies IdM cx0 , where IdM, cx0 N N by lemma

1. IdMog cx0 og implies g kx0 , where kx0 = cx0 og M N a constant map. Similarly IdMoh cx0 oh implies h kx0 . Transitivity of -homotopy gives g h, hence is universal relation. There fore [M, N] is singleton.

ACKNOWLEDGMENT

The research work of the first author is supported by the Council of Scientific and Industrial Research, India under grant award no: 09/1284(0001)2019-EMR-I with JRF exam roll no 400528.

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