Homotopy Self-Equivalences of ℝP2× S2 and a Braid for non-orientable $4$-Manifolds
Abstract
We compute the group of based homotopy classes of self-homotopy equivalences of the non-orientable $4$-manifold RP2× S2. It is an elementary abelian $2$-group of rank $5$. The subgroup of classes whose based lift to the universal cover S2× S2 has degree +1 has rank $4$ and is isomorphic to the group of free homotopy classes. The group maps onto the group of self-equivalences of the algebraic $2$-type, which has order $8$, with kernel π4(RP2× S2)≅(Z/2)2 generated by pinch-fold maps; that these maps are independent is shown by an adjunction argument. We then construct a non-orientable analogue of the Hambleton--Kreck braid of exact sequences in dimension $4$, for non-orientable manifolds with spin universal cover, stated for finite fundamental group. We apply the braid to RP2× S2 and to RP4, and for the Hambleton--Hillman manifold with fundamental group (Z/2)2 we determine the normal $1$-type and show that self-equivalences act on π2 through diagonal sign matrices. For RP2× S2 the pinch-fold maps are detected by twisted spin bordism, and, assuming facts from surgery theory stated as hypotheses, exactly one of the three non-trivial pinch-fold maps is homotopic to a homeomorphism.
Related
- Homotopy Self-Equivalences of Manifolds
- Disconnectedness of stability manifolds and their geometric loci, and separation of braid orbits of semiorthogonal decompositions
- Simple homotopy types of smooth four-manifolds with finite fundamental group
- Homotopy rigidity for quasitoric manifolds over a product of simplices
- Smooth manifolds homotopy equivalent to products of spheres
BibTeX
@misc{pamuk2026homotopy,
title = {{Homotopy Self-Equivalences of $\mathbbRP^2\times S^2$ and a Braid for non-orientable $4$-Manifolds}},
author = {Mehmetcik Pamuk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16606},
eprint = {2610.16606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16606},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Homotopy Self-Equivalences of ℝP^2× S^2 and a Braid for non-orientable $4$-Manifolds
AU - Mehmetcik Pamuk
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16606
UR - https://arxiv.org/abs/2610.16606
AB - We compute the group of based homotopy classes of self-homotopy equivalences of the non-orientable $4$-manifold RP^2× S^2. It is an elementary abelian $2$-group of rank $5$. The subgroup of classes whose based lift to the universal cover S^2× S^2 has degree +1 has rank $4$ and is isomorphic to the group of free homotopy classes. The group maps onto the group of self-equivalences of the algebraic $2$-type, which has order $8$, with kernel π_4(RP^2× S^2)≅(Z/2)^2 generated by pinch-fold maps; that these maps are independent is shown by an adjunction argument. We then construct a non-orientable analogue of the Hambleton--Kreck braid of exact sequences in dimension $4$, for non-orientable manifolds with spin universal cover, stated for finite fundamental group. We apply the braid to RP^2× S^2 and to RP^4, and for the Hambleton--Hillman manifold with fundamental group (Z/2)^2 we determine the normal $1$-type and show that self-equivalences act on π_2 through diagonal sign matrices. For RP^2× S^2 the pinch-fold maps are detected by twisted spin bordism, and, assuming facts from surgery theory stated as hypotheses, exactly one of the three non-trivial pinch-fold maps is homotopic to a homeomorphism.
ER -