Non-trivial smooth families of $K3$ surfaces

Baraglia, David

doi:10.48550/arxiv.2102.06354

Cited by 2 publications

(3 citation statements)

References 15 publications

(33 reference statements)

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“…Baraglia [3] and Lin [46] gave examples of smooth 1-connected 4-manifolds M for which π 2 (BDiff(M)) is not finitely generated, so the analogues of Theorems A and C fail in dimension 4. For 2n = 2, the result is well-known.…”

Section: Small Dimensionsmentioning

confidence: 99%

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Finiteness properties of automorphism spaces of manifolds with finite fundamental group

2023

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Given a closed smooth manifold M of even dimension $$2n\ge 6$$ 2 n ≥ 6 with finite fundamental group, we show that the classifying space $$B\textrm{Diff}(M)$$ B Diff ( M ) of the diffeomorphism group of M is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold $$N\subset M$$ N ⊂ M of arbitrary codimension into M has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group.

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Section: Small Dimensionsmentioning

confidence: 99%

“…Lemma 4. 3 Let M be a compact smooth manifold of dimension d ≥ 3 with π 1 (M) finite at all basepoints. Then the kth homotopy group of the total homotopy fibre of (22) is finitely generated for k ≥ 2 and finite for k = 1.…”

Section: The Case Of Finite Second Homotopy Groupmentioning

confidence: 99%

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

2023

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“…Small dimensions. Baraglia showed that π 2 (BDiff(M )) is not finitely generated for M a complex K3-surface [Bar21], so the analogues of Theorems A and C fail for closed simply connected 4-manifolds. It seems not to be known whether in this dimension Theorem C also fails in positive codimension.…”

Section: Infinite Fundamental Groupsmentioning

confidence: 99%

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Bustamante¹,

Krannich²,

Kupers³

2021

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Given a closed smooth manifold M of even dimension 2n ≥ 6 with finite fundamental group, we show that the classifying space BDiff(M ) of the diffeomorphism group of M is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold N ⊂ M of arbitrary codimension into M has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite.We begin with the main result.Theorem A. Let M be a closed smooth manifold of dimension 2n ≥ 6. If π 1 (M ) is finite at all basepoints, then the space BDiff(M ) and all its homotopy groups are of finite type.Here BDiff(M ) denotes the classifying space of the group Diff(M ) of diffeomorphisms of M in the smooth topology and being of finite type refers to the following finiteness condition: Definition. A space X is of finite type if it is weakly homotopy equivalent to a CW-complex with finitely many cells in each dimension. A group G is of finite type (or of type F ∞ ) if it admits an Eilenberg-MacLane space K(G, 1) of finite type.Theorem A has the following immediate corollary. Corollary B.For a manifold M as in Theorem A, the homotopy groups of BDiff(M ) are degreewise finitely generated. The same holds for the homology and cohomology groups with coefficients in any Z[π 0 (Diff(M ))]-module A that is finitely generated as an abelian group.Remark. There are variants of Theorem A for spaces of homeomorphisms and for manifolds M that have boundary or come with tangential structures (see Section 5).Before explaining further applications of Theorem A and elaborating on its assumptions, we put this result in a historical context. During this overview, we fix a closed connected smooth manifold M of dimension d ≥ 6.Context. The study of finiteness properties of diffeomorphism groups and their classifying spaces has a long history in geometric topology. Combining work Borel-Serre [BS73] and Sullivan [Sul77], it was known since the 70's that if M is simply connected, then the group π 0 (Diff(M )) of isotopy classes of diffeomorphisms of M is of finite type and thus finitely generated. It was also known that this result has its limits: based on work of Hatcher-Wagoner [HW73], Hatcher [Hat78, Theorem 4.1] and Hsiang-Sharpe [HS76, Theorem 2.5] showed that this can fail in the presence of infinite fundamental group, for instance for the high-dimensional torus M = × d S 1 . Later Triantafillou [Tri99, Corollary 5.3] weakened the condition on the fundamental group to allow all finite groups as long as M is orientable.These results can be interpreted as finiteness results for the fundamental group of the classifying space BDiff(M ) for smooth fibre bundles with fibre M . Around the same time as Sullivan's work appeared, Waldhausen [Wal78] developed a programme to systematically study these and related classifying spaces. It was known (see [Hat78, Proposition 7.5] for a proof outline) tha...

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Non-trivial smooth families of $K3$ surfaces

Cited by 2 publications

References 15 publications

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

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