Let $X$ be a connected space and $\Pi_1(X)$ be its fundamental groupoid. We consider the homologous relation $\mathcal R$ on every morphism space: $f,g\in \Pi_1(X)(p,q)$ are related if the singular one-chain $g-f= \partial S$ for a two-chain $S$.

It seems this is a congruence relation: if we further have $f'-g'=\partial S'$ for $f',g'\in \Pi_1(X)(q,r)$ then $f'\circ f-g'\circ g= \partial(S+S')$. So we can consider the quotient category $$ \Pi'_1(X):=\Pi_1(X)/\mathcal R $$ Is this stuff well-studied and is there a good reference? For example,

  1. It is known that the fundamental groupoid can be viewed as a functor $\Pi_1: \mathcal{Top}\to \mathcal{Grpd}$ from the category of topological spaces to the category of groupoids. Does this property also holds for our quotient $\Pi_1'$?
  2. The fundamental group $\pi_1(X,p)$ is the object group of $\Pi_1(X)$ at $p\in X$; in analogy, it seems $H_1(X)$ is simply the object group at any point $p\in X$ of our new $\Pi'_1(X)$. Is this correct?

The resulting groupoid is equivalent to the disjoint union of groupoids B(H_1(X_i)) taken over all connected components X_i of X. This answers both 1 and 2 in the positive.

To see this, observe that maps in both directions can be constructed using the corresponding universal properties. The fundamental group maps to the first homology group via the Hurewicz homomorphism. Vice versa, any 1-cycle can be converted to a continuous loop (in a nonunique way), and any two such choices will differ by a 2-boundary, which yields the desired map to the quotient under consideration.

  • $\begingroup$ Thank you! Let's say $X$ is connected. Could you explain the notation $B(H_1(X))$ or mention a reference? I am not an expert about this. $\endgroup$ – Hang Aug 25 at 22:53
  • 3
    $\begingroup$ For a group $G$, here $BG$ means the category with one object and with the hom given by $G$. $\endgroup$ – Todd Trimble Aug 25 at 22:54

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