Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer
    
Although I couldn't download it, it seems to be the paper I am looking for. Thank you. –  Niyazi Mar 6 '11 at 4:47

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.