Let $M^4$ be a closed orientable smooth manifold and $$ I: H_2(M,\mathbb Z) \times H_2(M,\mathbb Z) \to \mathbb Z $$ its intersection form. If $b^+ \ge 1$, where $b^+$ denotes the number of positive eigenvalues of $I$ induced on $H^2(M,\mathbb R)$, can we find a $c \in H_2(M,\mathbb Z)$ such that $$ I(c,c) >0. $$
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4$\begingroup$ Yes. There is an $a$ in $H_2(M,\mathbb{R})$ with $I(a,a)>0$, then any $b$ in $H_2(M,\mathbb{Q})$ sufficiently close to $a$ will have $I(b,b)>0$, and some multiple of $b$ will be in $H_2(M,\mathbb{Z})$. Note that this is a general fact about quadratic forms, nothing particular about the intersection form. $\endgroup$– abxCommented Oct 28, 2022 at 4:35
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