Understanding Quotient Spaces in Topology: Exploring Attachments and Retractions

[gazzo]

Let X,Y be two spaces, A a closed subset of X, f:A--->Y a continuous map. We denote by $X\cup_fY$ the quotient space of the disjoint union $X\oplus{Y}$ by the equivalence relation ~ generated by $a ~ f(a)$ for all a in A. This space is called teh attachment of X with Y along A via f.

i) If A is a strong deformation retract of X, show that Y is a deformation retract of $X\cup_fY$.

ii) The map f is extendable to a continuous map F:X-->Y if and only if Y is a retract of $X\cup_fY$.

I'm really stuck on this question. If A is a strong deformation retract of X, then we have some homotopy $H:X\times[0,1]\rightarrowX$ where for all x in X, H(x,0) = x. H(x,1) is in A. and for all b in A and all t in [0,1], H(a,t) = a.

But what has that got to do with $X\cup_fY$? :yuck:

Are the members of $X\cup_fY$ the equivalence classes where for a~b if one of: a = b, f(a)=f(b), a is in A and b=f(u) in Y.

So we want to come up with a map $\varphi:X\cup_fY\rightarrow{Y}$ where for all y in Y, $\varphi(y) = y$.

 

[matt grime]

Things in the attachment are things in X or Y but where we identify an element in A with its image. Example: X=Y= a disc A is the boundary of X and f is an isomorphism onto the boudary of Y. The attachment is then two discs identified along their boundary, ie a sphere.

now imagine X is a punctured disc, so it is strongly homotopic to its boundary, S^1. Then it is clear that the join along the boundary is now a punctured sphere, and that this is homotopy equivalent to the disc Y, and that this homotopy can just be taken by extending the strong homotopy of X to its boundary by making it the identity on Y.

 

[gazzo]

hmm thanks a lot Matt I think I've got it.

 

[matt grime]

topology is/was for me very frustrating. the results in it are easily seen to be true but it is always hard to write down the proofs because they are so fiddly. take for instance the proof that the fundamental group is indeed a group (associativity of loops) and independent of base point (in a path connected space). try to think of an easy example. at least you will be able to desribe that so that your teachers understand you understand what is going on.