Discrete homotopy and homology theories for finite posets

Abstract

This paper presents a discrete homotopy theory and a discrete homology theory for finite posets. In particular, the discrete and classical homotopy groups of finite posets are always isomorphic. Moreover, this discrete homology theory is related to the discrete homotopy theory through a discrete analogue of the Hurewicz map.

Figures (5)

• Figure 1: The Hasse diagram of $\mathbb{Z}^2$.
• Figure 2: The poset $X$ and the geometric realization $|\mathcal{K}(X)|$.
• Figure 3: $P_1\mathrel{\hbox{o}rigin=c]{-45}{$⇒$}} P_2$ is a sunny collapse.
• Figure 4: $P_2\mathrel{\hbox{o}rigin=c]{-45}{$⇒$}} P_3$ is not a sunny collapse.
• Figure 5: A poset $X$ and the geometric realization $|\mathcal{K}(X)|$.

Theorems & Definitions (52)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Lemma 3.4