Homotopy Self-Equivalences of Manifolds
Abstract
Hambleton and Kreck described the group of based homotopy self-equivalences of a closed oriented $4$-manifold by a commutative braid of exact sequences built from bordism groups and the Postnikov $2$-type. We construct the analogous braid for closed oriented BO⟨ n+1⟩-manifolds M of dimension 2n or 2n−1 whose fundamental group is finite or infinite cyclic and which satisfy one condition on the action of π1 on the homotopy of the fibre of M→ Pn(M). We compute Aut•(S1× Sm−1)≅(ℤ/2)2 for m≥4, and Aut•(S3× S3)≅(ℤ/12)2rtimes SL2(ℤ) with the standard action modulo $12$. Further, Aut•(S3× S4)≅(ℤ/2⊕ℤ/24)rtimes(ℤ/2)2 has order $192$, that Aut•(S4× S4)≅ D4×ℤ/2 has order $16$, and that for connected sums of r≥2 copies of S3× S3 the kernel contains a subgroup (ℤ/12)2r on which the permutations of the summands act non-trivially. For Sp× Sp the kernel is a sum of two copies of π2p(Sp) modulo Whitehead products with ιp, and the image is determined by an obstruction on the Whitehead square. In particular Aut•(S5× S5) is isomorphic to the theta subgroup of SL2(ℤ) and Aut•(S7× S7) is an extension of SL2(ℤ) by (ℤ/120)2.
Related
- Homotopy Self-Equivalences of ℝP2× S2 and a Braid for non-orientable $4$-Manifolds
- 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
- Rational homotopy theory of flag manifolds
BibTeX
@misc{pamuk2026homotopy,
title = {{Homotopy Self-Equivalences of Manifolds}},
author = {Mehmetcik Pamuk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02244},
eprint = {2610.02244},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02244},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Homotopy Self-Equivalences of Manifolds
AU - Mehmetcik Pamuk
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.02244
UR - https://arxiv.org/abs/2610.02244
AB - Hambleton and Kreck described the group of based homotopy self-equivalences of a closed oriented $4$-manifold by a commutative braid of exact sequences built from bordism groups and the Postnikov $2$-type. We construct the analogous braid for closed oriented BO⟨ n+1⟩-manifolds M of dimension 2n or 2n−1 whose fundamental group is finite or infinite cyclic and which satisfy one condition on the action of π_1 on the homotopy of the fibre of M→ P_n(M). We compute Aut_•(S^1× S^m−1)≅(ℤ/2)^2 for m≥4, and Aut_•(S^3× S^3)≅(ℤ/12)^2rtimes SL_2(ℤ) with the standard action modulo $12$. Further, Aut_•(S^3× S^4)≅(ℤ/2⊕ℤ/24)rtimes(ℤ/2)^2 has order $192$, that Aut_•(S^4× S^4)≅ D_4×ℤ/2 has order $16$, and that for connected sums of r≥2 copies of S^3× S^3 the kernel contains a subgroup (ℤ/12)^2r on which the permutations of the summands act non-trivially. For S^p× S^p the kernel is a sum of two copies of π_2p(S^p) modulo Whitehead products with ι_p, and the image is determined by an obstruction on the Whitehead square. In particular Aut_•(S^5× S^5) is isomorphic to the theta subgroup of SL_2(ℤ) and Aut_•(S^7× S^7) is an extension of SL_2(ℤ) by (ℤ/120)^2.
ER -