This must be an elementary topology question that ought to appear in the first few chapters of any topology book. Consider the isomorphisms of a topological space to itself. Each is a transformation and they thus form a group. Some such isomorphisms can be continuously deformed into others. This is an equivalence relation. The equivalence sets form a group. This group is rather fundamental to the space but is not the ‘fundamental group’. For instance this group for the torus ((S2)2) has eight elements.
I need the name of this group to explain some differences in various notions of the group of a topological fiber bundle. I would think that for each element in the group of a bundle, there is a loop in the base space for the bundle that causes that transformation on the fiber space. Most (all?) definitions seem to allow more group elements than this.
Perhaps it should be called the symmetry group of the topology. The reals: [0, 1] have a topological symmetry group of 2. The isomorphism x → x2 is equivalent to the identity. The isomorphism x → 1−x is not.
It seems clear to me that the group for the Möbius strip, as a bundle, should be the two group.
The automorphisms of a topological space, form a topological group. The connected components thereof form a group like the group we seek.
Perhaps the group should be called the topology’s discrete symmetry group.