Mapping class groups of highly connected $$(4k+2)$$-manifolds

Krannich, Manuel

doi:10.1007/s00029-020-00600-7

Cited by 17 publications

(23 citation statements)

References 54 publications

(88 reference statements)

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“…Remark 3.4 For a complete description of π 0 Diff D 2n (V g ), one still needs to determine the extension problems of the first or third part of the theorem, which we do not pursue at this point. Similar extensions by Kreck [26] describing the closely related mapping class group π 0 Diff ∂ (W g,1 ) for n ≥ 3 have been resolved in [25] for n odd.…”

Section: Remark 32supporting

confidence: 53%

A homological approach to pseudoisotopy theory. I

Krannich¹

2022

Invent. math.

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We construct a zig–zag from the once delooped space of pseudoisotopies of a closed 2n-disc to the once looped algebraic K-theory space of the integers and show that the maps involved are p-locally $$(2n-4)$$ ( 2 n - 4 ) -connected for $$n\,{>}\,3$$ n > 3 and large primes p. The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik–Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa’s stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification is a calculation of the rational homotopy groups of $$\mathrm {BDiff}_\partial (D^{2n+1})$$ BDiff ∂ ( D 2 n + 1 ) in degrees up to $$2n-5$$ 2 n - 5 .

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Section: Remark 32supporting

confidence: 53%

A homological approach to pseudoisotopy theory. I

Krannich¹

2022

Invent. math.

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“…For a complete description of π 0 Diff D 2n (V ), one still needs to determine the extension problems of the first or third part of the theorem, which we do not pursue at this point. Similar extensions by Kreck [Kre79] describing the closely related mapping class group π 0 Diff ∂ (W , 1 ) for n ≥ 3 have been resolved in [Kra19] for n odd.…”

supporting

confidence: 53%

A homological approach to pseudoisotopy theory. I

Krannich

2020

Preprint

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MANUEL KRANNICH A. We construct a zig-zag from the space of pseudoisotopies of a closed 2n-disc to the once looped algebraic K -theory space of the integers and show that the maps involved are p-locally (2n − 4)-connected for n > 3 and large primes p. The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik-Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa's stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification is a calculation of the rational homotopy groups of BDiff ∂ (D 2n+1 ) in degrees up to 2n − 5.

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“…Later Kreck determined the mapping class groups of almost parallelizable (n − 1)-connected 2n-manifolds up to extension problems ( [18]). Recent progresses include the study of the extension problems ( [19], [3], [16]) and the homological stablity of BDiff(♯g(S n × S n ), D) ( [7]). Other related results are for example the computation of the set of mapping degrees between (n − 1)connected 2n-manifolds ( [6], [5]) and the existence of almost complex structures ( [30]).…”

Section: Introduction To the Main Resultsmentioning

confidence: 99%

Free cyclic group actions on highly-connected $2n$-manifolds

Yang

2019

Preprint

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In this paper we study smooth orientation-preserving free actions of the cyclic group Z/m on a class of (n − 1)-connected 2n-manifolds, ♯g(S n × S n )♯Σ, where Σ is a homotopy 2n-sphere. When n = 2 we obtain a classification up to topological conjugation. When n = 3 we obtain a classification up to smooth conjugation. When n ≥ 4 we obtain a classification up to smooth conjugation when the prime factors of m are larger than a constant C(n).

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Mapping class groups of highly connected $$(4k+2)$$-manifolds

Abstract: We compute the mapping class group of the manifolds $$\sharp ^g(S^{2k+1}\times S^{2k+1})$$ ♯ g ( S 2 k + 1 × … Show more

Cited by 17 publications

References 54 publications

A homological approach to pseudoisotopy theory. I

A homological approach to pseudoisotopy theory. I

A homological approach to pseudoisotopy theory. I

Free cyclic group actions on highly-connected $2n$-manifolds

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