Is the mapping class group of
Is the mapping class group of
In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the
At least
The best references I know for the surgery structure set of
From this, it's not immediately clear to me whether enough is known to run Hatcher's argument for
Has anybody computed the various mapping class groups of
(I would also be interested in other infinite families of high-dimensional manifolds, like real projective spaces and lens spaces, but I imagine those require strictly more work to understand, given the presence of non-trivial fundamental group without being aspherical like the torus is.)
K2-Think
The mapping class group of complex projective space ( \mathbb{CP}^n ) for ( n \geq 3 ) is known in the smooth, PL, and topological categories. Here's a synthesis of the key points:
Homotopy Automorphism Group (( \text{hAut}(\mathbb{CP}^n) )):
- ( \pi_0(\text{hAut}(\mathbb{CP}^n)) \cong \mathbb{Z}/2 ), generated by the homotopy class of complex conjugation (which acts as ( -1 ) on (...
GPT-5
Short answer. Yes: for n ≥ 3 the mapping class group of CP^n (in each of the topological, PL and smooth categories) is Z/2, generated by complex conjugation. Equivalently:
- π0(Homeo(CP^n)) ≅ π0(PL(CP^n)) ≅ π0(Diff(CP^n)) ≅ Z/2.
- The nontrivial class is represented by complex conjugation [z0:…:zn] ↦ [z̄0:…:z̄n].
- Orientation: conjugation reverses orientation when n is odd and preserves it when n...
Claude Opus 4
This is a great question about the mapping class groups of complex projective spaces. Let me address what is known and what the challenges are.
What is known
The smooth/PL/topological mapping class groups of
- The diffeomorphism group structure: Kreck and Su showed in "A surgery sequen...
o3 Pro
Write Diff for the group of smooth diffeomorphisms of a manifold, PL-Diff for PL–homeomorphisms and Top-Diff for (purely) topological homeomorphisms, and write
π₀Diff(M)=Diff(M)/Diff₀(M)
for the corresponding mapping class group (isotopy classes of maps that preserve orientation unless the contrary is stated).
Throughout let n ≥ 3, so dim CPⁿ = 2n ≥ 6 and the usual surgery theory applies.
...
o4 mini
One can show by the usual “surgery‐on–simply‐connected even‐complexes” argument that nothing exotic ever appears on ℂPⁿ for n≥3, so in every category DIFF, PL or TOP the mapping‐class group is just the group of homotopy‐classes of self‐equivalences, namely ±1 on H². In particular:
- Surgery on compact simply‐connected manifolds with cohomology only in even degrees (Wall “Surgery on compact ma...
DeepSeek R1
First, I need to recall what mapping class groups are in different categories: smooth, PL, and topological. The user seems familiar with Hatcher's approach using surgery theory and homotopy ...
Claude 3.7 Sonnet
Mapping Class Groups of Complex Projective Spaces
The mapping class group of
Known Results
For
For $n...
Gemini 2.5 Pro
This is an excellent question that sits at the intersection of several deep areas of geometric topology. The short answer is:
**The mapping class groups of
o3
Let
Mod𝒞( CPⁿ )=π₀𝒞(CPⁿ) , 𝒞 = Diff , PL , Top
be the (smooth, PL, respectively topological) mapping–class group of the complex projective space of complex dimension n (real dimension 2n ≥6, i.e. n ≥ 3).
1 The two obvious classes of self-maps
Any self map induces a ring‐isomorphism
H²(CPⁿ;ℤ)=ℤ⟨x⟩ with x^{n+1}=0.
He...
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