Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 19th 2013
   • (edited Jul 19th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe I am being dumb, but help me anyway: where is a discussion of Poincaré duality genuinely on the level of chain complexes? Hence: characterizing Poincaré duality spaces $X$ by the fact that the chain complex $C_\bullet(X)$ is a dual object in the symmetric monoidal $(\infty,1)$-category of chain complexes? I am roughly aware of what's called "chain duality", L-theory and Ranicki's results. But none of what I see considers genuine dualizability of chain complexes in the $\infty$-category of chain complexes (or let it be the derived category, for starters). Can anyone help me? Sorry if this is too stupid. Just give me a pointer.

   Maybe I am being dumb, but help me anyway:

   where is a discussion of Poincaré duality genuinely on the level of chain complexes? Hence: characterizing Poincaré duality spaces $X$ by the fact that the chain complex $C_\bullet(X)$ is a dual object in the symmetric monoidal $(\infty,1)$-category of chain complexes?

   I am roughly aware of what’s called “chain duality”, L-theory and Ranicki’s results. But none of what I see considers genuine dualizability of chain complexes in the $\infty$-category of chain complexes (or let it be the derived category, for starters).

   Can anyone help me? Sorry if this is too stupid. Just give me a pointer.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 19th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexHm, maybe the result in * [[Thomas Tradler]], [[Mahmoud Zeinalian]], [[Dennis Sullivan]], _Infinity Structure of Poincare Duality Spaces_ ([arXiv:math/0309455](http://arxiv.org/abs/math/0309455)) gives this? Not sure yet.

   Hm, maybe the result in

   • Thomas Tradler, Mahmoud Zeinalian, Dennis Sullivan, Infinity Structure of Poincare Duality Spaces (arXiv:math/0309455)

   gives this? Not sure yet.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 19th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexah, got it! Yay. It appears as theorem, 2.5.2 of * Eric J. Malm, _String topology and the based loop space_ ([arXiv:1103.6198](http://arxiv.org/abs/1103.6198)) (well hidden, though, by the fact that the notation for chains is confusingly similar to that of homology! ;-/)

   ah, got it! Yay.

   It appears as theorem, 2.5.2 of

   • Eric J. Malm, String topology and the based loop space (arXiv:1103.6198)

   (well hidden, though, by the fact that the notation for chains is confusingly similar to that of homology! ;-/)