A combinatorial model for the Menger curve
Abstract
We represent the universal Menger curve as the topological realization |M| of the projective Fraïssé limit M of the class of all finite connected graphs. We show that M satisfies combinatorial analogues of the Mayer–Oversteegen–Tymchatyn homogeneity theorem and the Anderson–Wilson projective universality theorem. Our arguments involve only 0-dimensional topology and constructions on finite graphs. Using the topological realization M↦|M|, we transfer some of these properties to the Menger curve: we prove the approximate projective homogeneity theorem, recover Anderson's finite homogeneity theorem, and prove a variant of Anderson–Wilson's theorem. The finite homogeneity theorem is the first instance of an "injective" homogeneity theorem being proved using the projective Fraïssé method. We indicate how our approach to the Menger curve may extend to higher dimensions.
Additional Information
© 2022 World Scientific Publishing Co Pte Ltd. Research of Solecki supported by NSF grants DMS-1800680 and 1700426.Attached Files
Submitted - 1803.02516.pdf
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Additional details
- Eprint ID
- 114802
- Resolver ID
- CaltechAUTHORS:20220518-820731800
- DMS-1800680
- NSF
- DMS-1700426
- NSF
- Created
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2022-05-19Created from EPrint's datestamp field
- Updated
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2022-05-19Created from EPrint's last_modified field