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Day Convolution and Homotopy Groups on the Pointed Homotopy Category of CW Complexes
Abstract
We freely enrich the pointed homotopy category of CW complexes over abelian groups and equip the resulting functor category with the enriched Day convolution induced by the smash product. After identifying the convolution unit and recording the coherence maps, we compute the convolution of the homotopy-group functors. For integers n,m≥ 2 we construct a natural isomorphism πn⋆πm≅πn+m induced by the external smash product; equivalently, the external smash product is the universal natural pairing of homotopy groups. Under this identification, the symmetry interchanging the two factors acts on πn+m by multiplication by (−1)nm.
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BibTeX
@misc{yedathore2026convolution,
title = {{Day Convolution and Homotopy Groups on the Pointed Homotopy Category of CW Complexes}},
author = {Abhiram Srinivas Yedathore},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29239},
eprint = {2610.29239},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29239},
}
RIS
TY - UNPB
TI - Day Convolution and Homotopy Groups on the Pointed Homotopy Category of CW Complexes
AU - Abhiram Srinivas Yedathore
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29239
UR - https://arxiv.org/abs/2610.29239
AB - We freely enrich the pointed homotopy category of CW complexes over abelian groups and equip the resulting functor category with the enriched Day convolution induced by the smash product. After identifying the convolution unit and recording the coherence maps, we compute the convolution of the homotopy-group functors. For integers n,m≥ 2 we construct a natural isomorphism π_n⋆π_m≅π_n+m induced by the external smash product; equivalently, the external smash product is the universal natural pairing of homotopy groups. Under this identification, the symmetry interchanging the two factors acts on π_n+m by multiplication by (−1)^nm.
ER -