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
   • CommentTimeNov 3rd 2018
   • (edited Nov 3rd 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFor $X$ and $Y$ pointed topological spaces, the monoidalness of the suspension spectrum functor induces a canonical morphism of spectra $$ \Sigma^\infty Maps(X,Y) \longrightarrow Maps(\Sigma^\infty X, \Sigma^\infty Y) $$ Can we say anything useful about this transformation? For instance, may we get control over its cokernel, say when both $X$ and $Y$ are finite complexes with a connectivity bound on $Y$ but no dimension bound on $X$? (So Freudenthal suspension theorem fails, but can we quantify how much it fails, stably?)

   For $X$ and $Y$ pointed topological spaces, the monoidalness of the suspension spectrum functor induces a canonical morphism of spectra

   $$\Sigma^\infty Maps(X,Y) \longrightarrow Maps(\Sigma^\infty X, \Sigma^\infty Y)$$

   Can we say anything useful about this transformation? For instance, may we get control over its cokernel, say when both $X$ and $Y$ are finite complexes with a connectivity bound on $Y$ but no dimension bound on $X$? (So Freudenthal suspension theorem fails, but can we quantify how much it fails, stably?)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 6th 2018
   • (edited Nov 6th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI failed to notice the obvious, namely that the map in #1 is right away the projection to the first stage in the Goodwillie-Taylor tower of the functor $Maps(X,-)$ $$ \Sigma^\infty Maps(X, Y) \overset{p_1}{\longrightarrow} (P_1 Maps(X,-))(Y) = (D_1 Maps(X,-))(Y) = Maps( \Sigma^\infty X, \Sigma^\infty Y ) $$ Thanks to Charles Rezk over on the MO homotopy-chat for patiently pointing this out. I'd still like get a better idea of bounding $coker(\pi_0(p_1))$.

   I failed to notice the obvious, namely that the map in #1 is right away the projection to the first stage in the Goodwillie-Taylor tower of the functor $Maps(X,-)$

   $$\Sigma^\infty Maps(X, Y) \overset{p_1}{\longrightarrow} (P_1 Maps(X,-))(Y) = (D_1 Maps(X,-))(Y) = Maps( \Sigma^\infty X, \Sigma^\infty Y )$$

   Thanks to Charles Rezk over on the MO homotopy-chat for patiently pointing this out.

   I’d still like get a better idea of bounding $coker(\pi_0(p_1))$.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 6th 2018
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexPerhaps I'm being dense, but what is the $S^4$ doing there in #2?

   Perhaps I’m being dense, but what is the $S^4$ doing there in #2?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 6th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for catching. That's just the choice for $Y$ that I have in mind, of course. Fixed now.

   Thanks for catching. That’s just the choice for $Y$ that I have in mind, of course. Fixed now.