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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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Infinite homotopy stable class for 4-manifolds with boundary
We show that for every odd prime $$q$$, there exists an infinite family~$$\{M_i\}_{i=1}^{\infty}$$ of topological 4-manifolds that are all stably homeomorphic to one another, all the manifolds $$M_i$$ have isometric rank one equivariant intersection pairings and boundary $$L(2q, 1) \# (S^1 \times S^2)$$, but they are pairwise not homotopy equivalent via any homotopy equivalence that restricts to a homotopy equivalence of the boundary.
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- Award ID(s):
- 2347230
- PAR ID:
- 10500516
- Publisher / Repository:
- Mathematical Sciences Publishers
- Date Published:
- Journal Name:
- Pacific journal of mathematics
- ISSN:
- 0030-8730
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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