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](http://www.sciencedirect.com/science/article/pii/S0166864110003354), 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](http://www.sciencedirect.com/science/article/pii/S0166864110003354)). 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](http://arxiv.org/abs/math/9509228) 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!



Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$
My former doctoral student Lubomyr Zdomskyy has resolved this problem, noticing that adding $\omega\_2$ Cohen reals to a model of GCH produces a model in which the cardinal $\omega\_1$ admits a monotone cofinal map $\omega^\omega\to(\omega\_1)^{\omega\_1}$. 


Alternatively the same result can be derived from the existence of a K-Lusin set of cardinality $\mathfrak c=2^{\omega\_1}$ in $\omega^\omega$.


A subset $L$ of a Polish space $X$ is called a *K-Lusin set* in $X$ if for every compact subset $K\subset X$ the intersection $K\cap L$ is countable.


**Theorem** (Zdomskyy). If there exists a $K$-Lusin set $L\subset\omega^\omega$ of cardinality $|L|=\mathfrak c=2^{\omega\_1}$, then there exists a mootone cofinal map $f:\omega^\omega\to(\omega\_1)^{\omega\_1}$. 


*Proof*. Let $g:L\to (\omega\_1)^{\omega\_1}$ be any surjective map. Observe that for every $x\in\omega^\omega$ the set $L\_x=\{y\in L:y\le x\}$ is countable. Then the formula $f(x)=\sup(g(L\_x))$ defines a required cofinal monotone map $f:\omega^\omega\to(\omega\_1)^{\omega\_1}$. $\square$


The existence of a K-Lusin set of cardinality $\mathfrak c$ in the Cohen model was established by [Bartoszynski and Halbeisen](http://arxiv.org/pdf/math/0107165.pdf).


