For a fixed coefficient field \(F\), define the Poincaré series of a space \(X\) to be the formal power series \(p(t) = \sum_i a_i t^i\) where \(a_i\) is the dimension of \(H^i(X; F)\) as a vector space over \(F\), assuming this dimension is finite for all \(i\). Show that \(p(X \times Y) = p(X) p(Y)\). Compute the Poincaré series for \(S^n\), \(\mathbb{RP}^n\), \(\mathbb{RP}^\infty\), \(\mathbb{CP}^n\), \(\mathbb{CP}^\infty\), and the spaces in the preceding three exercises.
Solution
Over the fixed field \(F\), the Künneth theorem has no Tor term and gives \[ H^n(X\times Y;F)\cong\bigoplus_{i+j=n}H^i(X;F)\otimes_F H^j(Y;F). \] Taking dimensions therefore yields convolution of the Betti numbers, hence \[ \boxed{p_{X\times Y}(t)=p_X(t)p_Y(t).} \]
The basic examples are \[ p_{S^n}(t)=1+t^n. \] For \(\mathbb{RP}^n\) the answer depends on \(\operatorname{char}F\): \[ p_{\mathbb{RP}^n}(t)= \begin{cases} 1+t+\cdots+t^n,&\operatorname{char}F=2,\\ 1,&\operatorname{char}F\ne2,\ n\text{ even},\\ 1+t^n,&\operatorname{char}F\ne2,\ n\text{ odd}, \end{cases} \] and \[ p_{\mathbb{RP}^\infty}(t)= \begin{cases} (1-t)^{-1},&\operatorname{char}F=2,\\ 1,&\operatorname{char}F\ne2. \end{cases} \] For complex projective spaces, \[ p_{\mathbb{CP}^n}(t)=1+t^2+\cdots+t^{2n} =\frac{1-t^{2n+2}}{1-t^2}, \qquad p_{\mathbb{CP}^\infty}(t)=\frac1{1-t^2}. \]
For Exercise 12, both spaces have one copy of \(F\) in degree \(0\) and in every degree \(\ge2\) except degree \(1\), with the same ring structure; thus \[ p(t)=1+\frac{t^2}{1-t}. \] For Exercise 13, \[ \widetilde H^*(\mathbb{CP}^\infty/\mathbb{CP}^1;F) \] has one generator in every even degree \(\ge4\), so \[ p(t)=1+\frac{t^4}{1-t^2}. \] For Exercise 14, let \(X\) be the pushout built from \(\mathbb{RP}^\infty\) and \(\mathbb{CP}^n\). If \(\operatorname{char}F=2\), the ring description \[ H^*(X;F)\cong F[\alpha,\beta]/(\beta^2-\alpha^{2n+1}), \qquad |\alpha|=2,\ |\beta|=2n+1, \] shows that it is free as an \(F[\alpha]\)-module on \(1,\beta\). Hence \[ p_X(t)=\frac{1+t^{2n+1}}{1-t^2}. \] If \(\operatorname{char}F\ne2\), the \(\mathbb{RP}^\infty\) tail contributes no positive-dimensional cohomology, so \(X\) has the same \(F\)-cohomology as \(\mathbb{CP}^n\) and \[ p_X(t)=\frac{1-t^{2n+2}}{1-t^2}. \]
For the analogous pushout \(Y\) associated to \(\mathbb{CP}^\infty\to\mathbb{HP}^\infty\), the pair \((Y,\mathbb{HP}^n)\) is the same cellular pair as \((\mathbb{CP}^\infty,\mathbb{CP}^{2n+1})\). Thus, over every field, \[ H^*(Y;F) \] has one generator in degrees \(0,4,8,\ldots,4n\) and one generator in every even degree \(\ge4n+4\), with no other groups. Therefore \[ \boxed{ p_Y(t)=1+t^4+\cdots+t^{4n}+\frac{t^{4n+4}}{1-t^2}. } \] These formulas complete the Poincaré-series calculation for all spaces in Exercises 12–14.