The following questions should be very easy, but I need help. Notations are same as Bredon's geometry and topology book. This question is related to chapter VII.3.page 441. $\nabla \colon X\vee X\rightarrow X$ denotes the codiagonal, $S$ denotes the reduced suspension operation.

Q1:Is every example of H-cogroups given by the pair $(\nabla, S)?$ In other words, is it possible to obtain a H-cogroup which is not given by suspensions? Could you give me some references where they discuss these objects,play with $\nabla$ and change the fibers of the Serre fibrations accordingly?

Q2: Does it make sense to ask if the inclusion $J_f\subset S^n$ is a cofibration where $f\colon S^n\rightarrow S^n$ and $J_f$ is the Julia set of $f?$ (I should admit that I haven't spent time on this question yet, please skip it if it is too general)

Thank you.


Regarding question (1), the answer is that there are co-groups which aren't suspensions.

Berstein, Israel; Harper, John R. Cogroups which are not suspensions. Algebraic topology (Arcata, CA, 1986), 63–86, Lecture Notes in Math., 1370, Springer, Berlin, 1989.

A cogroup is a co-H-space with an associative comultiplication and an inversion. The paper gives the first examples of spaces which are cogroups but are not homotopy equivalent to a suspension. These examples are 2- and 3-cell complexes. The proofs involve delicate piecing together of homotopies and detailed information about homotopy groups of spheres.

  • $\begingroup$ Although I couldn't download it, it seems to be the paper I am looking for. Thank you. $\endgroup$ – Niyazi Mar 6 '11 at 4:47

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