Bounded-domain degree from relative homology

Let $d\ge 1$, let $U\subset {\mathbb{R}}^d$ be bounded and open, and let $f:\bar{U}\to {\mathbb{R}}^d$ be continuous. For $y\mathrm{\notin }f(\mathrm{\partial }U)$ the set $K=f^{-1}(\{y\})\cap U$ is compact in $U$: its closure preimage is closed in the compact set $\bar{U}$ and avoids $\mathrm{\partial }U$. Give $U$ its standard orientation. Let $[U{]}_K\in H_d(U,U\setminus K;\mathbb{Z})$ be the unique relative orientation class restricting to the standard local orientation generator at each $x\in K$; for $K=\mathrm{\varnothing }$ use the zero class. Let $[{\mathbb{R}}^d{]}_{\{y\}}$ be the positive generator of $H_d({\mathbb{R}}^d,{\mathbb{R}}^d\setminus \{y\};\mathbb{Z})\cong \mathbb{Z}$. The bounded-domain degree is the integer $\mathrm{deg}(f,U,y)$ determined by

$f_{\ast }[U{]}_K=\mathrm{deg}(f,U,y)[{\mathbb{R}}^d{]}_{\{y\}},$

where $f$ is regarded as a map of pairs $(U,U\setminus K)\to ({\mathbb{R}}^d,{\mathbb{R}}^d\setminus \{y\})$. The following local lemma proves that this definition is well-defined and has the required normalization, additivity, and moving-target homotopy properties.

Relative singular homology

R-orientation of a topological manifold

Local homology detects manifold dimension, interior, and boundary

Compatible orientation classes over compact subsets

Functoriality of relative homology

Chizat and Bach, Appendix C.3.1

Hatcher, Algebraic Topology, §2.2