Is there any intuition behind the definition of clutching datum?

Question

In Tom Dieck’s Algebraic Topology he defines in 1.3.6 the term “clutching datum” as following:

Let $(U_j| j\in J)$ be a family of sets. Assume that for each pair $(i, j)\in J\times J$ a subset $U_i^j\subset U_i$ is given as well as a bijection $g_i^j:U_i^j\to U_j^i$. We call the families $(U_j, U_j^k, g_j^k)$ a clutching datum if:

(1) $U_j=U_j^j$ and $g_j^j=\text{id}$.

For each triple $(i, j, k)\in J\times J\times J$ the map $g_i^j$ induces a bijection $$g_i^j: U_i^j\cap U_i^k\to U_j^i\cap U_j^k$$ and $g_j^k g_i^j=g_i^k$ holds, considered as maps from $U_i^j\cap U_i^k$ to $U_k^j \cap U_k^i$.

Answer 1

tom Dieck defines the concept of a clutching datun for families of sets, but the relevant application is for topological spaces. Here the subsets $U_i^j \subset U_i$ are assumed to be open and the bijections $g_i^j$ are assumed to be homeomorphims.

But let us consider the concept on the level of sets. The clutching datum generates an equivalence relation $\sim$ on the disjoint sum $\coprod_i U_i$ and thus yields a surjection $h : \coprod_i U_i \to X = \coprod_i U_i/\sim$. In the topological case $X$ is endowed with the quotient topology.

The equivalence relation identifies any two points with are connected by some of the bijections $g_i^j$. If you image each $U_i$ as a patch of fabric, then you sew together the patches as desribed by the $g_i^j$ to get $X$.

The $h_i = h \mid_{U_i} : U_i \to X$ are injective and the $U(i) = h(U_i) \subset X$ overlap. In fact, $U(i) \cap U(j) = h(U_i^j) = h(U_j^i)$.

Answer 2

Do you know the definition of an atlas of a differentiable manifold?

Think of the $U_j$'s as an atlas and the $g^j_i$'s as the overlap maps.

Now ask yourself the question: if a point $x$ in my manifold is represented by points in three different coordinate charts, say $x_i \in U_i$, $x_j \in U_j$, and $x_k \in U_k$, what can I derive about the overlap maps applied to these three representative points?