In their article [Instanton counting and Donaldson invariants](https://arxiv.org/abs/math/0606180) the authors define the slant product for $\beta \in H\_i(X)$ (where $X$ is a manifold) as following.


Let $P \to X$ and SO(3) bundle and $M(P)$ the moduli space of irreducible anti-self-dual connections. Let $\mathcal{P}$ be the universal bundle $X \times M(P)$, then the slant product is defined as:
$$
\mu(\beta) := -\frac{1}{4}p\_1(\mathcal{P})/\beta
$$
This in the end of page 5. **What does it even mean to divide by a homology class?** 



Gottsche Nakajima Yoshioka define a weird slant product
The slant product is defined in many basic algebraic topology books, eg Hatcher's. If you are willing to work over a field, then you'd take the Kunneth decomposition of $p\_1 = \sum x\_j \otimes m\_j$ with $x\_j \in H^\*(X)$ and $m\_j \in H^\*(M(P))$. Then $p\_1/\beta = \sum x\_j(\beta) \otimes m\_j \in H^\*(M(P))$, where the evaluation $x\_j(\beta)$ is declared to be $0$ unless $x\_j$ is in the same dimension as $\beta$. 


If you like to think in terms of de Rham cohomology, you are doing a partial integration of a form representing $p\_1$ over the cycle $\beta$ to get a $4-i$ form.


