##
**Domination between different products and finiteness of associated semi-norms.**
*(English)*
Zbl 1495.57017

Consider closed oriented manifolds of dimension \(n\). It is said that \(M\) dominates \(N\), and written \(M \geq N\) if there is a continuous map \(f : M \to N\) of non-zero degree, that is, \(f_{\ast}([M]) = \deg(f)[N]\), \(\deg(f) \neq 0\), in homology or, equivalently, \(f^{\ast}([\omega(N)]) = \deg(f)[\omega(M)]\), \(\deg(f) \neq 0\), in cohomology (where \(\omega(M) \in H^{n}(M)\) denotes the cohomological fundamental class of \(M\)).

According to the author, the following question (Question 1.3) was posed to him by M. Gromov: Let \(X_1 \times \cdots \times X_m\) be a Cartesian product of closed oriented manifolds of positive dimensions. Which other non-trivial products dominate \(X_1 \times \cdots \times X_m\)?

In the present paper, the author determines “all possible dominations between different products of manifolds when none of the factors of the codomain is dominated by products.” The results give partial answers to the above question of M. Gromov. Precisely, the author proves the following:

Theorem 1.4: Suppose that \(X_1 \times \cdots \times X_m\), \(Y_1 \times \cdots \times Y_r\) are closed oriented manifolds of positive dimensions such that \(X_1,\ldots,X_m\) are not dominated by non-trivial direct products and \(\dim(X_1 \times \cdots \times X_m) = \dim(Y_1 \times \cdots \times Y_r)\). Then \(Y_1 \times \cdots \times Y_r \geq X_1 \times \cdots \times X_m\) if and only if \(Y_i \geq X_{a_{i_1}} \times \cdots \times X_{a_{i_{\xi_i}}}\) for all \(i = 1, \ldots, l\), where \(\xi_i \geq 1\), \(a_{ij} \in \{1,\ldots,m\}\), and \(a_{ij} \neq a_{i'j'}\) if \((i,j) \neq (i',j')\).

The proof of Theorem 1.4 is based on Thom’s work [R. Thom, Comment. Math. Helv. 28, 17–86 (1954; Zbl 0057.15502)] on the Steenrod problem [S. Eilenberg, Ann. Math. (2) 50, 247–260 (1949; Zbl 0034.25304), Problem 25] about realizing homology classes by closed manifolds.

A finite functorial semi-norm in degree \(k \in \mathbb N\) singular homology is a semi-norm \(\nu: H_k(X;\mathbb R) \to [0;1)\) for every topological space \(X\), where “functorial” means that the semi-norm \(\nu\) is not increasing under induced homomorphisms \(f : H_{\ast}(Y) \to H_{\ast}(X)\) for all continuous maps \(f : Y \to X\) [D. Crowley and C. Löh, Algebr. Geom. Topol. 15, No. 3, 1453–1499 (2015; Zbl 1391.57008); M. Gromov, Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. Boston, MA: Birkhäuser (1999; Zbl 0953.53002)]. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds.

In the current paper the author also proves the finiteness of every product-associated (for the definition, see the paper) functorial semi-norm on the fundamental classes of the products as above.

According to the author, the following question (Question 1.3) was posed to him by M. Gromov: Let \(X_1 \times \cdots \times X_m\) be a Cartesian product of closed oriented manifolds of positive dimensions. Which other non-trivial products dominate \(X_1 \times \cdots \times X_m\)?

In the present paper, the author determines “all possible dominations between different products of manifolds when none of the factors of the codomain is dominated by products.” The results give partial answers to the above question of M. Gromov. Precisely, the author proves the following:

Theorem 1.4: Suppose that \(X_1 \times \cdots \times X_m\), \(Y_1 \times \cdots \times Y_r\) are closed oriented manifolds of positive dimensions such that \(X_1,\ldots,X_m\) are not dominated by non-trivial direct products and \(\dim(X_1 \times \cdots \times X_m) = \dim(Y_1 \times \cdots \times Y_r)\). Then \(Y_1 \times \cdots \times Y_r \geq X_1 \times \cdots \times X_m\) if and only if \(Y_i \geq X_{a_{i_1}} \times \cdots \times X_{a_{i_{\xi_i}}}\) for all \(i = 1, \ldots, l\), where \(\xi_i \geq 1\), \(a_{ij} \in \{1,\ldots,m\}\), and \(a_{ij} \neq a_{i'j'}\) if \((i,j) \neq (i',j')\).

The proof of Theorem 1.4 is based on Thom’s work [R. Thom, Comment. Math. Helv. 28, 17–86 (1954; Zbl 0057.15502)] on the Steenrod problem [S. Eilenberg, Ann. Math. (2) 50, 247–260 (1949; Zbl 0034.25304), Problem 25] about realizing homology classes by closed manifolds.

A finite functorial semi-norm in degree \(k \in \mathbb N\) singular homology is a semi-norm \(\nu: H_k(X;\mathbb R) \to [0;1)\) for every topological space \(X\), where “functorial” means that the semi-norm \(\nu\) is not increasing under induced homomorphisms \(f : H_{\ast}(Y) \to H_{\ast}(X)\) for all continuous maps \(f : Y \to X\) [D. Crowley and C. Löh, Algebr. Geom. Topol. 15, No. 3, 1453–1499 (2015; Zbl 1391.57008); M. Gromov, Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. Boston, MA: Birkhäuser (1999; Zbl 0953.53002)]. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds.

In the current paper the author also proves the finiteness of every product-associated (for the definition, see the paper) functorial semi-norm on the fundamental classes of the products as above.

Reviewer: Danuta Kołodziejczyk (Warszawa)

### MSC:

57N65 | Algebraic topology of manifolds |

55M25 | Degree, winding number |

46B20 | Geometry and structure of normed linear spaces |

### References:

[1] | S. Eilenberg, On the problems of topology,Ann. of Math. (2)50(2)(1949), 247-260.DOI: 10.2307/1969448. · Zbl 0034.25304 |

[2] | M. Gromov,“Metric Structures for Riemannian and Non-Riemannian Spaces”, Based on the 1981 French original, With appendices by M. Katz, P. Pansu, and S. Semmes, Translated from the French by S. M. Bates, Progress in Mathematics152, Birkh¨auser Boston, Inc., Boston, MA, 1999. |

[3] | D. Kotschick and C. L¨oh, Fundamental classes not representable by products, J. Lond. Math. Soc. (2)79(3)(2009), 545-561.DOI: 10.1112/jlms/jdn089. · Zbl 1168.53024 |

[4] | D. Kotschick and C. L¨oh, Groups not presentable by products,Groups Geom. Dyn.7(1)(2013), 181-204.DOI: 10.4171/GGD/180. · Zbl 1286.20051 |

[5] | D. Kotschick, C. L¨oh, and C. Neofytidis, On stability of non-domination under taking products,Proc. Amer. Math. Soc.144(6)(2016), 2705-2710. DOI: 10.1090/proc/13043. · Zbl 1348.57033 |

[6] | D. Kotschick and C. Neofytidis, On three-manifolds dominated by circle bundles,Math. Z.274(1-2)(2013), 21-32.DOI: 10.1007/s00209-012-1055-3. · Zbl 1277.57003 |

[7] | C. Neofytidis, Fundamental groups of aspherical manifolds and maps of non-zero degree,Groups Geom. Dyn.12(2)(2018), 637-677.DOI: 10.4171/GGD/451. · Zbl 1483.57024 |

[8] | R. |

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