Let $ X $, $ Y $, and $ A $ be based connected CW-complexes; let $ Z $ be an $ H $-space; and let $ p:X\times Y\to A $ be a based map. We study generalized co-Gottlieb sets associated with the product domain $ X\times Y $ by means of separable maps. We show that the canonical restriction map $ i^\sharp:[X\times Y, Z]\longrightarrow[X\vee Y, Z] $ is bijective if and only if every element of $ [X\times Y, Z] $ is separable. For a grouplike $ H $-space $ Z $, this condition is equivalent to $ [X\wedge Y, Z] = 0 $. We also introduce a smash-product obstruction that measures the failure of separability and obtain an injectivity criterion for the restricted map $ \widetilde{i}^{\, \sharp}: p^\top(X\times Y, Z) \longrightarrow (p\circ i)^\top(X\vee Y, Z) $ without assuming $ [X\wedge Y, Z] = 0 $. Under the separability condition, we obtain natural coordinate injections. Finally, if $ Z $ is a grouplike $ H $-space and $ (p\circ i)^\top(X\vee Y, Z) $ is a finite cyclic group of prime-power order, then $ p^\top(X\times Y, Z) $ is a group.
Citation: Howon Choi. Generalized co-Gottlieb sets on product spaces via separable maps[J]. AIMS Mathematics, 2026, 11(9): 28849-28863. doi: 10.3934/math.20261148
Let $ X $, $ Y $, and $ A $ be based connected CW-complexes; let $ Z $ be an $ H $-space; and let $ p:X\times Y\to A $ be a based map. We study generalized co-Gottlieb sets associated with the product domain $ X\times Y $ by means of separable maps. We show that the canonical restriction map $ i^\sharp:[X\times Y, Z]\longrightarrow[X\vee Y, Z] $ is bijective if and only if every element of $ [X\times Y, Z] $ is separable. For a grouplike $ H $-space $ Z $, this condition is equivalent to $ [X\wedge Y, Z] = 0 $. We also introduce a smash-product obstruction that measures the failure of separability and obtain an injectivity criterion for the restricted map $ \widetilde{i}^{\, \sharp}: p^\top(X\times Y, Z) \longrightarrow (p\circ i)^\top(X\vee Y, Z) $ without assuming $ [X\wedge Y, Z] = 0 $. Under the separability condition, we obtain natural coordinate injections. Finally, if $ Z $ is a grouplike $ H $-space and $ (p\circ i)^\top(X\vee Y, Z) $ is a finite cyclic group of prime-power order, then $ p^\top(X\times Y, Z) $ is a group.
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Howon Choi. Generalized co-Gottlieb sets on product spaces via separable maps[J]. AIMS Mathematics, 2026, 11(9): 28849-28863. doi: 10.3934/math.20261148
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