Is there a compact $n$-dimensional manifold $M$ or, more generaly, a compact $n$-dimensional topological space $M$ with the following property?


"For every continuous map $f:M \to \mathbb{R}^{n}$ there are points $a,b,c \in M $ with $f(a)=f(b)=f(c)$."


This is motivated by the following obvious consequence of the Borsuk-Ulam theorem:


"For every continuous map $f:S^n \to \mathbb{R}^n$ there are points $a, b \in S^n$ with $f(a)=f(b)$." 



A generalization of the Borsuk Ulam theorem
If M is allowed to be a simplicial complex, five $n$-simplices with a common $(n-1)$-dimensional face should do the job. If $M$ should be a manifold, take any closed non-orientable one.


The multiplicity of maps between manifolds can be studied with the help of characteristic classes, see [this preprint of Roman Karasev](https://arxiv.org/abs/1002.0660) and [this preprint of Roman Karasev and Pavle Blagojevic](https://arxiv.org/abs/1603.06723). For example, any continuous map $\mathbb{R}\mathrm{P}^4 \to \mathbb{R}^4$ sends some four points in $\mathbb{R}\mathrm{P}^4$ to the same point in $\mathbb{R}^4$.


Another generalization of the Borsuk-Ulam theorem (more close to it in the spirit) deals with spaces with a group action (like $\mathbb{Z}\_2$-action on the sphere) such that for any map to $\mathbb{R}^n$ some orbit is sent to a point. This is discussed in detail in Section 6 of


*Matoušek, Jiří*, [**Using the Borsuk-Ulam theorem. Lectures on topological methods in combinatorics and geometry. Written in cooperation with Anders Björner and Günter M. Ziegler**](http://dx.doi.org/10.1007/978-3-540-76649-0), Universitext. Berlin: Springer (ISBN 978-3-540-00362-5/pbk; 978-3-540-76649-0/ebook). xii, 214~p. (2008). [ZBL1234.05002](https://zbmath.org/?q=an:1234.05002).


