I am now assuming that the answer to my problem is that since h is continuous (and carries x0 to y0 ) that h0f is a loop in Y
Clarification thanks to Opalg
(SOLVED)
On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachment giving Munkres pages 333-334)
"Suppose that is a continuous map that carries the point of X to the point of Y.
We denote this fact by writing:
If f is a loop in X based at , then the composite is a loop in Y based at "
I am confused as to how this works ... can someone help with the formal mechanics of this.
To illustrate my confusion, consider the following ( see my diagram and text in atttachment "Diagram ..." )
Consider a point that is mapped by f into i.e.
Then we would imagine that is mapped by into some corresponding point ( see my diagram and text in atttachment "Diagram ..." )
i.e.
BUT
But (see above) we only know of h that it maps into ? {seems to me that is not all we need to know about h???}
Can anyone please clarify this situation - preferably formally and explicitly?
Peter
I am now assuming that the answer to my problem is that since h is continuous (and carries x0 to y0 ) that h0f is a loop in Y
Clarification thanks to Opalg
(SOLVED)
remember, a loop at x_{0} in X is a (continuous) map f:I→X with:
f(0) = f(1) = x_{0}.
so it follows that a loop at y_{0} in Y is a map (in topology, by "map" one typically means continuous by assumption) g:I→Y with:
g(0) = g(1) = y_{0}.
so all that is happening here is we are taking g = hof, where h is a continuous map with h(x_{0}) = y_{0}
(that is, h is an arrow in the category of "pointed spaces" (topological spaces with a distingushed point, usually written as (X,x_{0}), Top_{*}:
h: (X,x_{0})→(Y,y_{0}).).
g is continuous, being the composition of two continuous maps.
g has domain I (since f has domain I), and co-domain Y (since h has co-domain Y).
furtherfore g(0) = hof(0) = h(f(0)) = h(x_{0}) = y_{0} = h(x_{0}) = h(f(1)) = hof(1) = g(1).
the whole point of this, is that we are going to create another category: Toph_{*}, which has the same set of objects (pointed spaces) that we map between, but our "arrows" (morphisms) are now going to be "base-point-preserving" homotopy equivalence classes [h]: (X,x_{0})→(Y,y_{0}).
in other words, all continuous maps that map x_{0} to y_{0} that can be continuously deformed to each other are going to be thought of as "essentially the same". in this view, the map (function) h of pointed spaces induces a "loop homotopy" [h], by:
[h]o[f] = [hof].
(although a loop is often "paramaterized by the unit interval", that is, we consider the base space (domain) to be (I,0), you can see how we could consider it to be the pointed space (S^{1},(1,0)) via the parameterization f:[0,1]→S^{1}, f(t) = (cos(2πt), sin(2πt)). this is useful, because later, when we consider the "loop product" [a]*[b], we can draw homotopy equivalences on the unit square. another way we can leverage this, is by using the natural complex multiplication of the circle in the complex plane, to define "point-wise" multiplication of loops, rather than a "concatenation product". the two different kinds of loop products yield "different loops", in general, but the two resulting loops are homotopic).
for example, it turns out that for loops a,b,c: (a*b)*c ≠ a*(b*c), but that ([a]*[b])*[c] = [a]*([b]*[c]). the following diagram is often used in the proof:
now the interesting thing about all this, is given two pointed spaces, (X,x_{0}) and (Y,y_{0}) with a base-point preserving map h between them, we can associate with each space a loop group, π_{1}(X,x_{0}), and π_{1}(Y,y_{0}), whose elements are homotopy equivalence classes of loops at x_{0},y_{0} respectively. this gives a functor from the category Top_{*} to the category Grp, π_{1}, which acts on both objects (pointed spaces) AND arrows (base-point preserving continuous maps), in this case π_{1}(h) = h_{*}, which is a homomorphism of groups (group homomorphisms are the "arrows" of the category Grp). given a second mapping, k: (Y,y_{0})→(Z,z_{0}), you might want to convice yourself that:
(koh)_{*} = k_{*}oh_{*}
we can also regard it as a functor from Toph_{*}→Grp, because if [h_{1}] = [h_{2}] (that is, if h_{1},h_{2} are homotopically equivalent), then (h_{1})_{*} = (h_{2})_{*}, so h_{*} only depends on [h].
this gives an easy way to prove R and S^{1} are not homeomorphic. if they were, then their fundamental groups would be isomorphic (say, choose 0 as a base point for R, and (1,0) as a base point for S^{1}). but π_{1}(R) ≅ {0}, while π_{1}(S^{1}) ≅ Z. we've found an "algebraic" way to find "holes".
this "switching of realms" isn't perfect, the fundamental group doesn't capture "all" of the information about a pointed space (a category theoist would remark that as a functor, π_{1} is neither full nor faithful). but it allows us to begin a (coarse) classification of topological spaces based on algebraic considerations. and groups (especially abelian groups) are considerably easier to classify than topological spaces, which can be downright bizarre.
Thanks that is really significantly helpful ... even though I have only glanced through it ...
Will now work through this in detail ...
Just in passing ... you point out that ... ... this is helpful to me in another issue regarding retractions and induced homomorphisms ... so it is all very helpful ..
Thanks for explicit formal presentation of the maths ... rather than vague verbal statements.
Also thanks for the examples ... I wish textbooks did more of this ...
Very helpful to those working alone ... gives me the confidence to move on in algebraic topology ... thanks
Peter
You make use of caegory theory in your post.
Do you think that category theory is helpful for those at undergraduate levels of math - perhaps senior undergraduate levels - looks as if it may well be helpful!
What reference do you recommend for those seeking a first basic understanding of category theory? [Possibly one that gives copious examples ... :-) ]
Peter
just a note to the (SOLVED) assertion of mine ... I now have an intuitive sense that a continuous mapping that carries into actually produces a corresponding loop in Y ... ...
HOWEVER ... ... I have been unable to go past the 'hand waving' stage and give a rigorous formal proof of this fact based only on the topological definition of continuity.
Can anyone help me with this?
Peter
the way i think of a topology is this: it is a set, with a notion of "nearness" imposed upon it (a point x is near a set A, if every neighborhood of x intersects A). obviously, exactly "what" we mean by "near" is going to depend on how we define a neighborhood. if a neighborhood is an epsilon-ball, we get our intuitive idea of "near" as "within distance epsilon". but different metrics, give different shaped "equidistant sets" (a sphere for the euclidean metric, an n-cube for the "box metric", etc.). and we might not have even a metric, but just some vague sense/criterion of "nearness". for example, in the indiscrete topology, "everything is near", while in the discrete topology "everything is far".
so what a continuous function does, is preserve "nearness". folding is OK, tearing or cutting, is not. a continuous function f takes neighbors of x to neighbors of f(x). this is what we mean when way say f^{-1}(U) is open, if U is open. so intuitively you can see that a loop in X gets mapped by h:X→Y to a loop in Y if h maps the "basepoint" (the start and end of the loop) of the loop in X, to the "basepoint" of the loop in Y.
but again, rigourously, it is just as i said in my earlier post.
let g = hof, where h(x_{0}) = y_{0} is continuous, and f:I→X is a loop at x_{0}, so that f(0) = f(1) = x_{0}.
then g(0) = hof(0) = h(f(0)) = h(x_{0}) = y_{0}.
and g(1) = hof(1) = h(f(1)) = h(x_{0}) y_{0}.
so g:I→Y, with g(0) = g(1) = y_{0}, that is g = hof is a loop in Y at y_{0}. that's all there is to it.
i'm not expert on category theory...i've picked up bits and pieces here and there. as i understand it, one of its main virtues is as a language for talking about "meta-structures", that is: constructions in mathematics that occur in various forms in different kinds of things. in a sense it's an alternative to set theory: with set theory, sets are in a sense "atomic", everything else (functions, operations, etc.) is a certain KIND of set, usually defined by a (first or second-order) logical formula, like:
A U B = {x in U: x in A or x in B}, which can be further broken down to something like:
(x ∈ A U B) ↔ ([(x ∈ A → x ∈ U) & (x ∈ B → x ∈ U)] & [(x∈ A) v (x∈ B)]), where U is in this case some "universe" set.
in category theory, the "atoms" are objects and arrows, which need not be sets and functions (although that is one important example). the collections of objects and arrows need not be sets, either, although in most cases (to avoid certain logical paradoxes), one considers categories in which the collections of objects and arrows ARE sets (these categories are called "small"), or where the collection of arrows between any two objects is a set (these are called "locally small").
to give an example of a "categorical construction": in many categories, one can make a new object P, from two objects A,B with the following property (called a "universal" property):
if there are two morphisms (what an morphism is varies from category to category, it in the category of sets, it is a function):
p_{1}: P→A
p_{2}: P→B
with the property that for any object C and mappings f:C→A, g:C→A there is a UNIQUE mapping k:C→P such that:
f = p_{1}ok
g = p_{2}ok
then P is called the product of A and B.
in the category of sets, P is the cartesian product AxB, and the functions p_{1},p_{2} are the projection functions: p_{1}(a,b) = a, p_{2}(a,b) = b. in the category of groups, P is the direct product of A and B, and the projection homomorphisms are the same (this shows that AxB/(A x {e}) is isomorphic to B, since the projection homomorphisms are surjective). in the category of F-vector spaces, P is the direct sum of the vector spaces A and B, in the category of topological spaces, P is the product space of A and B (this "forces our hand" in defining the product topology, as we must have the projection maps continuous).
the standard reference for category theory is:
Categories For the Working Mathematician, by Saunders MacLane. this text assume a good deal of familiarity with undergraduate mathematics.
i find this a bit easier to read:
Arrows, Structures and Functors, by Michael Arbib.
John Armstrong has a nice bit of material on his blog: The Unapologetic Mathematician
there is also a brief introduction to category theory in:
Basic Algebra II, by Nathan Jacobson
that said, the motivation of using category theory comes mainly from algebraic topology, so learning it "in and of itself" may not make a whole lot of sense.