

Important Spaces and Operation on Them

In topology, we study
topological spaces
and continuous maps between them. Which spaces should
we begin to study? There are many wierd or
pathological spaces, which are chosen as examples in
textbooks to show the intuition of the reader is
false.
What Poincaré wanted to study in
“Analysis Situs”
[
Poi96
]
is the topology of
manifolds
. The category of manifolds is not a good place to
play. Possible operations are quite limited. In fact,
Poincaré’s ﬁrst attempt to
deﬁne the “homology group” as the
set of the equivalence classes of submanifolds under
the relation “homologous” by taking the
union of two manifolds and to deﬁne the
addition in homology by taking unions of submanifolds
caused a lot of discussions and criticism. The union
of submanifolds is not necessarily a manifold.
In order to overcome this difficulty, Poincaré
used triangulations. Namely he deﬁned homology
groups for
simplicial complexes
. The world of simplicial complexes is not large
enough. We usually work in the category of
CWcomplexes
or
simplicial sets
. And there are some more attempts to construct
convinient categories for doing algebraic topology
.


