On locally 1-connectedness of quotient spaces and its applications to fundamental groups
Let be a locally 1-connected metric space and be connected, locally path connected and compact pairwise disjoint subspaces of . In this paper, we show that the quotient space obtained from by collapsing each of the sets ’s to a point, is also locally 1-connected. Moreover, we prove that the induced continuous homomorphism of quasitopological fundamental groups is surjective. Finally, we give some applications to find out some properties of the fundamental group of the quotient space .
affil1]Ali Pakdaman affil2]Hamid Torabi affil2]Behrooz Mashayekhy
1 Introduction and Motivation
Let be an equivalent relation on a topological space . Then one can consider the quotient topological space and the quotient map . In general, quotient spaces are not well behaved and it seems interesting to determine which topological properties of the space may be transferred to the quotient space . For example, a quotient space of a simply connected or contractible space need not share those properties. Also, a quotient space of a locally compact space need not be locally compact.
Let be a finite collection of pairwise disjoint subspaces of a topological space . Then by we mean the quotient space obtained from by identifying each of the sets ’s to a point. The main result of this paper is as follows:
Theorem A. If is a locally 1-connected metric space and are connected, locally path connected and compact pairwise disjoint subspaces of , then the quotient space is locally 1-connected.
The quasitopological fundamental group of a based space is the familiar fundamental group endowed with the
quotient topology with respect to the canonical function identifying
path components on the loop space with the compact-open topology (see Br ()).
It is known that this construction gives rise a homotopy invariant functor from the homotopy category of based spaces to the category of quasitopological group and continuous homomorphism Br1 (). By applying the above functor on the quotient map , we have a continuous homomorphism . Recently, the authors T () proved that if is a first countable, connected, locally path connected space and are disjoint path connected, closed subspaces of , then the image of is dense in . By using the results of T () and Theorem A we have the primary application of the main result of this paper as follows:
Theorem B. If is a locally 1-connected metric space and are disjoint connected, locally path connected and compact subspaces of , then for each the continuous homomorphism
is an epimorphism.
Also, by some examples, we show that is not necessarily onto if the hypotheses of the above theorem do not hold.
Finally, we give some applications of the above results to explore the properties of the fundamental group of the quotient space . In particular, we prove that if is a separable, connected, locally 1-connected metric space and are connected, locally path connected and compact subsets of , then is countable. Moreover, is finitely presented. Note that with the recent assumptions on and the ’s, is simply connected when is simply connected.
2 Notations and preliminaries
For a pointed topological space , by a path we mean a continuous map . The points and are called the initial point and the terminal point of , respectively. A loop is a path with . For a path , denotes a path such that , for all . Denote by , two paths with the same initial and terminal points are called homotopic relative to end points if there exists a continuous map such that
The homotopy class containing a path is denoted by . Since most of the homotopies that appear in this paper have this property and end points are the same, we drop the term “relative homotopy” for simplicity. For paths with , denotes the concatenation of and , that is, a path from to such that , for and , for .
For a space , let be the space of based maps from to with the compact-open topology. A subbase for this topology consists of neighborhoods of the form , where is compact and is open in . We will consistently denote the constant path at by . The quasitopological fundamental group of a pointed space may be described as the usual fundamental group with the quotient topology with respect to the canonical map identifying homotopy classes of loops, denoted by . A basic account of quasitopological fundamental groups may be found in Br (), Br1 () and C1 (). For undefined notation, see M ().
A () A quasitopological group is a group with a topology such that inversion , , is continuous and multiplication is continuous in each variable. A morphism of quasitopological groups is a continuous homomorphism.
Br1 () is a functor from the homotopy category of based topological spaces to the category of quasitopological groups.
A space is semi-locally simply connected if for each point , there is an open neighborhood of such that the inclusion induces the zero map or equivalently a loop in can be contracted inside .
Br1 () Let be a connected and locally path connected topological space. The quasitopological fundamental group is discrete for some if and only if X is semi-locally simply connected.
T () Let be disjoint path connected, closed subsets of a first countable, connected, locally path connected space such that is semi-locally simply connected, then for each , is an epimorphism.
In this note all homotopies are relative. Also, when is a path, for brevity by we mean where is a suitable linear homeomorphism.
3 Constructing homotopies
This section is dedicated to some technical lemmas about homotopy properties of loops.
Assume that is a path in and are real numbers such that . Let , for . Then we know that ((M, , Theorem 51.3))
Let be a path in and be closed disjoint subintervals of . If and are homotopic relative to end points for , then , where
If is a subset of a topological space , then we denote the complement of in by .
Let be a loop in based at . Let be an increasing sequence such that and form a nested neighborhood basis at such that for every integer . If is a path from to such that is a null loop in , for every integer , then , where
It is obvious that , for every integers and hence it suffices to prove . Let by . Since is compact and is an open neighborhood of , there exist such that , where . Let be the path defined by . By definition of and since for , by a homotopy in . Therefore
by a homotopy in
and hence there exists a continuous map such that
(i) for ,
(ii) for ,
(iii) for ,
Since is compact and is an open neighborhood of , there exist such that , where . Let be the path defined by . Then is null in by the homotopy (note that is homeomorphic to ). Also, is null in because and hence by homotopy . By the similar way, for every we have the path by for which is null in by a homotopy , where and (see Figure 1).
Then and . Let and be an open neighborhood of . There exist such that . By the construction of ’s, there exist such that for each which implies that for each . Hence is continuous(preserving the other convergent sequences by is obvious).
In the following examples we show that the above lemma does not hold if is only null in .
Let be the Griffiths’ space: the one-point union of two copies of the cone over the Hawaiian earring as described in Gr (). Let ’s and ’s be the consecutive loops of the right and left Hawaiian earring, respectively. Then all loops and are null by homotopies in . Let , for every and define by
where ’s are suitable linear homeomorphisms from to . Note that is not null in (S, , Example 2.5.18). Let , and . Then is homotopic to a null loop or (depended on is even or odd). Also, every loop defined as like as in Lemma 3.3 is null and hence is not homotopic to , for every .
Let be a loop in based at , be closed and form a nested neighborhood basis at . If and by a homotopy in (), then is nullhomotopic.
Let and . Assume by a homotopy .
Since and , it remains to show that is continuous. Let . If , then , for a and every . Continuity of implies that . If , then the construction of , ’s and nested property of ’s imply that , for an open neighborhood of and sufficiently large . By the next example we show that Lemma 3.4 does not hold if is a null loop in instead of .
Let , , and be defined as in Example 3.3 and be the common point of two cones. We have all the hypothesis of the lemma except and are null in and is not null.
4 The main result
This section dedicated to prove Theorem by introducing notions -contained for subspaces of a given space and nested basis for a subspace .
Let be a topological space, be path connected subset of . We say that is -contained in at if for every , where are inclusion maps. Equivalently, every loop in is homotopic to a loop in by a relative homotopy in .
For a topological space and open subsets , we say is simply connected in if every loop in is nullhomotopic in . If is a subset of a topological space , then we denote the boundary of in by . Also, in a metric space , by we mean the open ball .
Let be a metric space and be a locally path connected, closed subset of . Let be an open neighborhood of which is simply connected in an open subset of . Then there exists an open neighborhood of the point such that and is -contained in at .
Since is metric, there exists such that . Since is locally path connected, there exists a neighborhood of such that is path connected. There exists such that the closed ball is contained in . Let be the open ball , then we show that is -contained in at . Let be a loop in with base point in . If is a component of , then . Since is path connected, there exists a path in from to .
Let . If for every , is a path from to , then by a homotopy in since is a loop in . Using Corollary 3.1, is homotopic to a loop in by a homotopy in .
Let . Since is uniformly continuous, there exists such that if , then . Also, there exists such that for every , . Similar to Case 1, there are paths such that , for by homotopies in . Therefore is homotopic to a loop by a homotopy in and .
Let and . If and , then by definition of and , exist and which implies that . Since is path connected, there exists a path with and such that by a homotopy in because is a loop in . Now, let and . Similarly, and there exists a path with and such that by a homotopy in because is a loop in . Since is compact and , there exists such that . Hence, we can replace , by the path , for , with homotopies in . Therefore is homotopic to a loop in by a homotopy in , as desired.
For a topological space , a family of open neighborhoods of a subset is called a nested basis of if for all and for every open set containing , there is such that . Let a nested basis of be a refinement of a nested basis of i.e for each . Then we say that is -refinement in at if is -contained in at for each .
There is two natural questions that whether a subset of a topological space has a nested basis and whether every nested basis has a -refinement? Obviously, compact subsets of metric spaces have nested basis and for the second question we have the following theorem. We recall that a topological space is said to be locally 1-connected if it is locally path connected and locally simply connected.
Let be a locally 1-connected metric space and be a locally path connected compact subset of . Then every nested basis of has a -refinement in at .
Let be the open ball with radius of , then is a nested basis for . Since is locally 1-connected, for every and there is an open neighborhood at such that is simply connected in . By Lemma 4.2, there exists such that is -contained in at . By local 0-connectivity of we can assume every is path connected. Since is compact, for every there exists and such that .
We claim for every there exists integer such that for every and every couple with nonempty intersection, is -contained in at . By contradiction assume there exists no such for an , then there are a sequence , a couple and loops such that there is no loop in homotopic to by a homotopy in . Let . Since is compact, there is a subsequence converging to . Let be a simply connected neighborhood of in . By Lemma 4.2 there exists an open neighborhood such that is -contained in at . There exists such that for each , which implies , for . Therefore is homotopic to a loop in by a homotopy in which is a contradiction.
Let . We show that is -contained in at . Consider as a loop at . Since , there is the Lebesque number for this open cover. Let such that , then there is such that . Let , where is linear homeomorphism for . Since and , if is a path from to (’s and are path connected), then is a loop in which implies that is homotopic to a loop by a homotopy in . Also, there exists such that is homotopic to a loop in by a homotopy in and similarly there are ’s and ’s for such that
which implies that is homotopic to a loop in by a homotopy in , as desired. Note that if is not decreasing, then we can put be the path component of that contains .
Let be an arbitrary nested basis of . Since is a nested basis of , there exists a subsequence of such that for every . Let be the -refinement of . Then is a -refinement of , where for each .
If is a metric space with a metric , then for the quotient space , the map defined by