Definition. A CW-complex A can be elementary contracted to a CW-complex B if there is a deformation retraction $A \to B$ or a quotient map by a contractible subcopmlex $A \to B$ (meaning according to a subcomplex in any cellular structure, i.e., the cell structure can be varied)
Definition. Contractibility is the transitive closure of "elementary contractility".
It is clear that contractility is a preorder (any homeomorphism is an elementary contraction). The minimal elements in it will be called homotopy incompressible.
- It is clear that among the 1-dimensional CW-complexes these are exactly bouquets of circles, and all the other one-dimensional ones are deformation retracted on them.
Questions:
Is it true that $S^n$ are homotopically incompressible? It is known that no subspace $\mathbb{R}^n$ is equivalent to $S^n$, therefore they have no proper deformation retracts. Intuitively, it seems that factorization over any contractible subcomplex will leave a sphere a sphere, but I still don't understand how to prove this.
In general, closed manifolds have no proper deformation retracts. Are they incompressible?
Is it possible to classify homotopically incompressible 2-dimensional CW-complexes?
Is it true that each CW-complex has a homotopically incompressible deformation retract (i.e., do the minimal elements in our order form a barrier from below / are there any chains going as many downward as desired)?
Due to (3), it is interesting - is it true that every connected n-dimensional CW-complex is homotopy equivalent to a bunch of n-dimensional closed manifolds? What is the criterion for incompressibility for non-closed manifolds? Are homotopy incompressible spaces closed with respect to a bouquet, product ..
Update 1: of course, the homotopy types of CW-complexes are not limited to bouquets of closed n-manifolds since the latter have a restriction on n-dimensional homology (first comment).
Update 2: After commenting on the error in the first statement, I extended the contraction class.