Given a cardinal $\kappa$ consider the set $\kappa^\kappa$ of all functions from $\kappa$ to $\kappa$, endowed with the partial order $f\le g$ iff $f(\alpha)\le g(\alpha)$ for all $\alpha\in\kappa$.

A map $f:P\to Q$ between two partially ordered sets is called

*monotone*if for any points $x\le y$ in $P$ we get $f(x)\le f(y)$ in $Q$;*cofinal*if for any $q\in Q$ there is $p\in P$ with $f(p)\ge q$.

**Problem 1.** Is there an uncountable regular cardinal $\kappa$ admitting a monotone cofinal map $f:\omega^\omega\to\kappa^\kappa$?

**Remark.** It can be shown that such cardinal $\kappa$ belongs to the half-interval $[\mathfrak b,\mathfrak d)$, where $\mathfrak b$ and $\mathfrak d$ are the boundedness and dominating numbers of the poset $\omega^\omega$.

Therefore, under $\mathfrak b=\mathfrak d$ the answer to the above problem is "No".

**Is the same "No" true in ZFC?**

This question can be reformulated in topological terms as follows. Following Cascales, Orihuela and Tkachuk, we say that a topological space $X$ is *strongly $\omega^\omega$-dominated* if there exists a family $(K_\alpha)_{\alpha\in\omega^\omega}$ of compact subsets of $X$ such that $K_\alpha\subset K_\beta$ for every $\alpha\le\beta$ in $\omega^\omega$ and for every compact set $K\subset X$ there exists $\alpha\in\omega^\omega$ such that $K\subset K_\alpha$.

It is easy to see that Problem 1 is equivalent to the topological

**Problem 2.** Is the space $\kappa^\kappa$ strongly $\omega^\omega$-dominated for some uncountable cardinal $\kappa$?

Here $\kappa^\kappa$ carries the Tychonoff product topology and $\kappa$ is endowed with the standard order topology.

**A Known Related Fact** (Cascales-Orihuela-Tkachuk). For any uncountable cardinal $\kappa$ the space $\omega^\kappa$ is not strongly $\omega^\omega$-dominated and hence does not admit a monotone cofinal map $f:\omega^\omega\to\omega^\kappa$.

Concerning the consistency strength of the statements in Problems 1,2, let us recall the following result of Cummings and Shelah. For a cardinal $\kappa$ let $\mathfrak b_\kappa$ and $\mathfrak d_\kappa$ be the boundedness and dominating numbers of $\kappa^\kappa$ endowed with the partial preorder $f\le^* g$ iff $|\{\alpha\in\kappa: f(\alpha)>g(\alpha)\}|<\kappa$.

**Theorem.** Let $F$ be a class function assigning to each regular cardinal $\kappa$ a triple of cardinals $(\beta(\kappa),\delta(\kappa),\lambda(\kappa))$ such that
$cf(\lambda(\kappa))>\kappa<\beta(\kappa)=cf(\beta(\kappa))\le\delta(\kappa)\le\lambda(\kappa)$ and $\lambda(\kappa)\le\lambda(\kappa')$ for any regular cardinals $\kappa<\kappa'$. Then there exists a class forcing poset $\mathbb P$ preserving all cardinals and cofinalities such that in the generic extension, $\mathfrak b_\kappa=\beta(\kappa)$, $\mathfrak d_\kappa=\delta(\kappa)$ and $|\kappa^\kappa|=\lambda(\kappa)$.

This result of Cumming and Shelah implies the consistency of $\mathfrak b=\omega_1$, $\mathfrak d=\omega_2$ and $\mathfrak b_\kappa=\mathfrak d_\kappa>\omega_2$ for all regular uncountable cardinals $\kappa$.

This impies that the consistency of the answer "No" to Problems 1,2 is strictly weaker that the consistency of the equality $\mathfrak b=\mathfrak d$.

On the other hand, the same result of Cummings and Shelah implies the consistency of $\mathfrak b=\omega_1$ and $\mathfrak b_{\omega_1}=\mathfrak d_{\omega_1}=\mathfrak d=\omega_1^{\omega_1}=\omega_2$, witnessing that a (desirable for me negative answer) to Problem 1 should use hardly the monotonicity (not only cofinality) of the map $f:\omega^\omega\to\kappa^\kappa$.

I need this "No" ZFC-answer (very much) for studying free topological groups. Please, help!