The Homotopy Type of the Loops on \((n-1)\)-Connected \((2n+1)\)-Manifolds

  • Samik BasuEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


For \(n\ge 2\), we compute the homotopy groups of \((n-1)\)-connected closed manifolds of dimension \((2n+1)\). Away from the finite set of primes dividing the order of the torsion subgroup in homology, the p-local homotopy groups of M are determined by the rank of the free Abelian part of the homology. Moreover, we show that these p-local homotopy groups can be expressed as a direct sum of p-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.


Homotopy groups Koszul duality Loop space Moore conjecture Quadratic algebra 

1991 Mathematics Subject Classification

Primary: 55P35 55Q52 Secondary: 16S37 57N15 



The author would like to thank the referee for pointing out the reference [7], and also for pointing out the history of the problem of loop space decompositions of highly connected manifolds.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Stat-Math Unit, Indian Statistical InstituteKolkataIndia

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