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There are several closely related concepts on the symmetries or symmetry groups of the space.

My apology, but some vague imprecise definitions may be as:

  1. Mapping class group (MCG) is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to the symmetries of the space.

  2. Diffeotopy group: The diffeotopy group D(M) of a smooth manifold M is the quotient of the diffeomorphism group Diff(M) by its normal subgroup Diff$_0$(M) of diffeomorphisms isotopic to the identity. Alternatively, one may think of the diffeotopy group as the group $π_0$(Diff(M)) of path components of Diff(M), since any continuous path in Diff(M) can be approximated by a smooth one, i.e. an isotopy.

  3. Isometry group: the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function.

These concepts are important and relevant for exotic $\mathbb{R}^4$, for example, we can ask: "Does every isometry group of an exotic $\mathbb{R}^4$ inject into its diffeotopy group?"

So my question here is aimed for a relation and interpretations between the definitions of Diffeotopy group, Mapping Class group, Isometry group?

For example, it seems that $$ \text{MCG($S^2 \times S^1)= \mathbb Z_2 \oplus \mathbb Z_2$} $$ $$ \text{extended-MCG($S^2 \times S^1)=\mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2$} $$ similarly,

$$ \text{ diffeotopy group of $(S^2 \times S^1)=\mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2$} $$

Are there general relations between the above Diffeotopy group, Mapping Class group, Isometry group?

wonderich
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    Unless $M$ is a surface, there is no standard definition of a mapping class group of $M$. You should spell out what definition you have in mind. – Moishe Kohan Mar 04 '19 at 01:11
  • Just a small nitpick — the mapping class group may not always be discrete. For finite-type surfaces this is true, but it is not for infinite-type surfaces. – Santana Afton Mar 04 '19 at 01:42
  • @Santana Afton, can you give examples? for finite-type surfaces and infinite-type surfaces??? – wonderich Mar 04 '19 at 03:49
  • A finite-type surface is one whose fundamental group is finitely generated: an example would be any compact surface with finite genus. An infinite-type surface is just ... not finite-type. An example is $\mathbb{C}-\mathbb{Z}$, or the Loch Ness Monster surface. – Santana Afton Mar 04 '19 at 03:55
  • If you want to see why these surfaces don’t have discrete mapping class group, consider the Loch Ness Monster surface, and note that you can find countably many disjoint curves on the surface. Then, take a sequence of multi twists about more and more of these curves! This sequence limits to the multi twists about each curve. – Santana Afton Mar 04 '19 at 04:00

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