Skip to main content
MathOverflow

Return to Question

added 26 characters in body
Source Link
Patricia Hersh
Patricia Hersh
  • 3.5k
  • 1
  • 30
  • 34

Recently, I've been wondering to what extent certain types of pathologies can arise in finite CW complexes -- notice that I do not want to assume that I'm in the PL category or that the CW complexes are regular, but do want to insist on there only being finitely many cells.

Specific question: is there a finite CW complex homeomorphic to a sphere such that one of its maximal cells has as its closure a ball withwhose boundary is embedded in the CW-sphere as an Alexander horned sphere as its boundary?

Follow-up question: if the answer is no, is there a finite CW complex homeomorphic to a sphere such that the closure of one of the maximal cells is a ball, but the closure of its complement is not a ball?

I don't have a specific reason I need to know this, but recently have been working a lot with finite CW complexes that are not in the PL category, or at least not obviously so, and would like to understand better the interaction between the finiteness of the number of open cells and the potential weirdness of the attaching maps. Information about other pathologies that can or cannot occur in this setting would also be much appreciated.

Thanks in advance for your help with this!

Recently, I've been wondering to what extent certain types of pathologies can arise in finite CW complexes -- notice that I do not want to assume that I'm in the PL category or that the CW complexes are regular, but do want to insist on there only being finitely many cells.

Specific question: is there a finite CW complex homeomorphic to a sphere such that one of its maximal cells has as its closure a ball with the Alexander horned sphere as its boundary?

Follow-up question: if the answer is no, is there a finite CW complex homeomorphic to a sphere such that the closure of one of the maximal cells is a ball, but the closure of its complement is not a ball?

I don't have a specific reason I need to know this, but recently have been working a lot with finite CW complexes that are not in the PL category, or at least not obviously so and would like to understand better the interaction between the finiteness of the number of open cells and the potential weirdness of the attaching maps. Information about other pathologies that can or cannot occur in this setting would also be much appreciated.

Thanks in advance for your help with this!

Recently, I've been wondering to what extent certain types of pathologies can arise in finite CW complexes -- notice that I do not want to assume that I'm in the PL category or that the CW complexes are regular, but do want to insist on there only being finitely many cells.

Specific question: is there a finite CW complex homeomorphic to a sphere such that one of its maximal cells has as its closure a ball whose boundary is embedded in the CW-sphere as an Alexander horned sphere?

Follow-up question: if the answer is no, is there a finite CW complex homeomorphic to a sphere such that the closure of one of the maximal cells is a ball, but the closure of its complement is not a ball?

I don't have a specific reason I need to know this, but recently have been working a lot with finite CW complexes that are not in the PL category, or at least not obviously so, and would like to understand better the interaction between the finiteness of the number of open cells and the potential weirdness of the attaching maps. Information about other pathologies that can or cannot occur in this setting would also be much appreciated.

Thanks in advance for your help with this!

Source Link
Patricia Hersh
Patricia Hersh
  • 3.5k
  • 1
  • 30
  • 34

Can the Alexander horned sphere arise as a cell boundary in a finite CW-sphere?

Recently, I've been wondering to what extent certain types of pathologies can arise in finite CW complexes -- notice that I do not want to assume that I'm in the PL category or that the CW complexes are regular, but do want to insist on there only being finitely many cells.

Specific question: is there a finite CW complex homeomorphic to a sphere such that one of its maximal cells has as its closure a ball with the Alexander horned sphere as its boundary?

Follow-up question: if the answer is no, is there a finite CW complex homeomorphic to a sphere such that the closure of one of the maximal cells is a ball, but the closure of its complement is not a ball?

I don't have a specific reason I need to know this, but recently have been working a lot with finite CW complexes that are not in the PL category, or at least not obviously so and would like to understand better the interaction between the finiteness of the number of open cells and the potential weirdness of the attaching maps. Information about other pathologies that can or cannot occur in this setting would also be much appreciated.

Thanks in advance for your help with this!