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We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to $$S^{2}$$ but do not admit a spine (that is, a piecewise linear embedding of $$S^{2}$$ that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer $$d$$ invariants.
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Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds
.We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Z n grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π 1 -injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated.
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- Award ID(s):
- 2105451
- PAR ID:
- 10626792
- Publisher / Repository:
- de Gruyter
- Date Published:
- Journal Name:
- Journal für die reine und angewandte Mathematik (Crelles Journal)
- Volume:
- 0
- Issue:
- 0
- ISSN:
- 0075-4102
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/Crelle-2024-0003
