Let $X$ be a surface (real, not complex). With coefficients in a ring $R$, there is an intersection pairing $\langle -,- \rangle:H_1(X;\mathbb R) \otimes H_1(X;\mathbb R) \to R$. Is this pairing perfect, or nondegenerate? I tend to get the terminology confused, but what I want to know is if the induced map $H_1(X) \to \operatorname{Hom}(H_1(X),R)$ is an isomorphism.

If $X$ is a closed surface, then my understanding is that the answer is yes. This is some guise of Poincare duality. But what if $X$ is noncompact, even infinite type (so that $H_1$ is not finite rank)? In particular, I'm curious about the case where $X$ is the infinite genus surface with one end. Here's a picture of that surface in the wild:

Figure: Hernández, Jesús Hernández; Morales, Israel; Valdez, Ferrán, The Alexander method for infinite-type surfaces, Mich. Math. J. 68, No. 4, 743-753 (2019). ZBL1481.57038.

Let $X$ be a surface (real, not complex). With coefficients in a ring $R$, there is an intersection pairing $\langle -,- \rangle:H_1(X;\mathbb R) \otimes H_1(X;\mathbb R) \to R$. Is this pairing perfect, or nondegenerate? I tend to get the terminology confused, but what I want to know is if the induced map $H_1(X) \to \operatorname{Hom}(H_1(X),R)$ is an isomorphism.

If $X$ is a closed surface, then my understanding is that the answer is yes. This is some guise of Poincare duality. But what if $X$ is noncompact, even infinite type (so that $H_1$ is not finite rank)? In particular, I'm curious about the case where $X$ is the infinite genus surface with one end. Here's a picture of that surface in the wild:

Figure: Hernández, Jesús Hernández; Morales, Israel; Valdez, Ferrán, The Alexander method for infinite-type surfaces, Mich. Math. J. 68, No. 4, 743-753 (2019). ZBL1481.57038.