8.4 Fiber sequences and the long exact sequence
If the codomain of a function is equipped with a basepoint , then we refer to the fiber of over as the fiber of . (If is connected, then is determined up to mere equivalence; see \autorefex:unique-fiber.) We now show that if is also pointed and preserves basepoints, then there is a relation between the homotopy groups of , , and in the form of a long exact sequence. We derive this by way of the fiber sequence associated to such an .
A pointed map between pointed types and is a map together with a path .
Given a pointed map between pointed types , we define a pointed map by
The path , which exhibits as a pointed map, is the obvious path of type
There is another functor on pointed maps, which takes to . When is pointed, we always consider to be pointed with basepoint , in which case is also a pointed map, with witness . Thus, this operation can be iterated.
Thus, any adjacent pair of maps in this fiber sequence is of the form
In particular, we have . We now observe that the types occurring in this sequence are the iterated loop spaces of the base space , the total space , and the fiber , and similarly for the maps.
For 1, we have
|(as is contractible)|
Tracing through, we see that this equivalence sends to , while its inverse sends to . In particular, the basepoint of is sent to , which equals . Hence this equivalence is a pointed map (see \autorefex:pointed-equivalences). Moreover, under this equivalence, is identified with .
Thus, the fiber sequence of can be pictured as:
where the minus signs denote composition with path inversion . Note that by \autorefex:ap-path-inversion, we have
Thus, there are minus signs on the -fold loop maps whenever is odd.
From this fiber sequence we will deduce an exact sequence of pointed sets. Let and be sets and a function, and recall from \autorefdefn:modal-image the definition of the image , which can be regarded as a subset of :
If and are moreover pointed with basepoints and , and is a pointed map, we define the kernel of to be the following subset of :
Of course, this is just the fiber of over the basepoint ; it a subset of because is a set.
Note that any group is a pointed set, with its unit element as basepoint, and any group homomorphism is a pointed map. In this case, the kernel and image agree with the usual notions from group theory.
An exact sequence of pointed sets is a (possibly bounded) sequence of pointed sets and pointed maps:
such that for every , the image of is equal, as a subset of , to the kernel of . In other words, for all we have
where denotes the basepoint of .
Usually, most or all of the pointed sets in an exact sequence are groups, and often abelian groups. When we speak of an exact sequence of groups, it is assumed moreover that the maps are group homomorphisms and not just pointed maps.
Let be a pointed map between pointed spaces with fiber . Then we have the following long exact sequence, which consists of groups except for the last three terms, and abelian groups except for the last six.
We begin by showing that the 0-truncation of a fiber sequence is an exact sequence of pointed sets. Thus, we need to show that for any adjacent pair of maps in a fiber sequence:
with , the sequence
is exact, i.e. that and .
The first inclusion is equivalent to , which holds by functoriality of and the fact that . For the second, we assume and and show there merely exists such that . Since our goal is a mere proposition, we can assume that is of the form for some . Now by \autorefthm:path-truncation, yields , so by a further truncation induction we may assume some . But now we have whose image under is , as desired.
Thus, applying to the fiber sequence of , we obtain a long exact sequence involving the pointed sets , , and in the desired order. And of course, is a group for , being the 0-truncation of a loop space, and an abelian group for by the Eckmann–Hilton argument (\autorefthm:EckmannHilton). Moreover, \autorefthm:fiber-of-the-fiber allows us to identify the maps and in this exact sequence as and respectively.
More generally, every map in this long exact sequence except the last three is of the form or for some . In the former case it is a group homomorphism, while in the latter case it is a homomorphism if the groups are abelian; otherwise it is an “anti-homomorphism”. However, the kernel and image of a group homomorphism are unchanged when we replace it by its negative, and hence so is the exactness of any sequence involving it. Thus, we can modify our long exact sequence to obtain one involving and directly and in which all the maps are group homomorphisms (except the last three). ∎
The usual properties of exact sequences of abelian groups can be proved as usual. In particular we have:
As an immediate application, we can now quantify in what way -connectedness of a map is stronger than inducing an equivalence on -truncations.
Let be -connected and , and define . Then:
If , then is an isomorphism.
If , then is surjective.
In \autorefsec:whitehead we will see that the converse of \autorefthm:conn-pik also holds.
|Title||8.4 Fiber sequences and the long exact sequence|