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1 Introduction
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This page is based on [Ranicki2002]. A map between manifolds represents a homology class . Let be an oriented cover with covering map . If factors through as then represents a homology class . Note that a choice of lift is required in order to represent a homology class.
Let be a basepoint of . By covering space theory (c.f. [Hatcher2002, Proposition 1.33]) a map can be lifted to if and only if , i.e. if and only if the composition is trivial for the quotient map.
The group is well-defined for any choice of lift since is a regular covering and changing the basepoint in to a different lift corresponds to conjugating by some .
2 Definition
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Let be an -dimensional manifold and let be an oriented cover. A -trivial map is a map from an oriented manifold with basepoint such that the composite
is trivial, together with a choice of lift .
3 Properties
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A map
that factors through
must map all of
to the same sheet of
, hence the pullback satisfies
Choosing where to lift a single point determines a lift
, which thought of as a map from
extends equivariantly to a lift
.
4 Lifts and paths - two alternative perspectives
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Rather than taking a lift as part of the data for a
-trivial map we could instead take an equivalence class of paths in
as is explained in this section. Since
is the group of deck transformations of
, the set of lifts
is non-canonically isomorphic to
with the group structure determined by the action of
once a choice of lift
has been chosen to represent the identity element. In this way the choice of lift that is included as part of the data of a
-trivial map can be thought of as a choice of isomorphism
Let
be a basepoint of
. The set of homotopy classes of paths from
to
is non-canonically isomophic to
. An isomorphism is defined by a choice of path
to represent the identity element:
where
denotes concatenation of paths and
is the path
in reverse. Let
be a basepoint of
that is a lift of
.
Define an equivalence relation
on this set by saying
The above isomorphism given by choosing
descends to give an isomorphism
where we use the same choice of path
to identify
with
.
Thus a choice of lift
corresponds to a choice of homotopy class of paths from
to
modulo
. A choice of lift
defines a bijection of sets
as follows. Given a choice of lift
choose any path
from
to
. Take the equivalence class of
which is a path in
from
to
. Conversely given a choice of class
choose any representative
. This lifts uniquely to a path
starting at
. Define a lift
by setting
. Note this map is well-defined since different choices of representative
may differ by elements of
but their lifts will still end at the same point.
To sum up we have the following diagram of non-canonical isomorphisms and bijections
Each map is obtained by making a choice and any two choices uniquely determine the third with the diagram commuting, so with two choices made the horizontal bijection is in fact an isomorphism of groups.
5 Examples
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...
6 References