# Is every polynomial with integral coefficients a Poincaré polynomial of a manifold?

For any compact, smooth, oriented manifold $$X$$ of dimension $$n$$ we can define its Poincaré polynomial $$p_X(z)=\sum_{k\geq0}b_k(X)z^k\in \mathbb Z[z],$$ which is the generating function of Betti numbers $$b_k(X)=\operatorname{rank} H_k(X)\in \mathbb N_0$$. Let $$q(z)=\sum_{k\geq0}c_kz^k$$ be a polynomial with nonnegative integral coefficients $$c_k\in\mathbb N_0$$, satisfying the Poincaré duality condition $$c_k=c_{n-k}$$. Does there exist a manifold $$X$$ with $$p_X=q$$?

• Another necessary condition: In dimension $4k+2$, the intersection pairing on $H^{2k+1}$ is non-degenerate and antisymmetric. Thus, $b_{2k+1}$ must be even if $n = 4k+2$. – Jason DeVito Apr 8 at 16:51

Here is the answer for connected (smooth, closed, oriented) 4-manifolds: Every symmetric polynomial of the form $$q(z)= 1+ a z+ b z^2 + a z^3 + z^4$$ is realized as the Poincaré polynomial of some 4-manifold, provided that $$a, b\in {\mathbb N}_0$$. Indeed, take $$M_{a,b}$$ equal to the connected sum of $$a$$ copies of $$S^3\times S^1$$ and $$b$$ copies of $$CP^2$$. For the computation of homology under connected sum, see this question. The end result is that $$b_i(M\# N)=b_i(M)+ b_i(N)$$ unless $$i=0$$ or $$i=d$$, where $$M, N$$ are both $$d$$-dimensional closed oriented manifolds. (The answer given there implicitly assumes that the manifolds are closed, i.e. compact and with empty boundary.)

More generally, you can also prescribe the constant (= the highest) coefficient of the Poincaré polynomial as long as they are $$=n\in {\mathbb N}$$, by taking the disjoint union of $$M_{a,b}$$ with $$n-1$$ copies of $$S^4$$.

• Amazing, thank you! – El Rafu Apr 8 at 20:07

Any connected, compact, orientable 2-dimensional manifold is a genus $$g$$ surface. Its Poincaré polynomial is $$1+2gz+z^2$$. So quadratic polynomials where the coefficient of $$z$$ is odd cannot be the Poincaré polynomial of any compact, orienteable 2-manifold.

• I see, thanks! Are you aware of such constraints for instance in $d=4$? – El Rafu Apr 7 at 14:51