Dual of an iterated mapping cylinder

1.1. Let \(\Delta^n = \{(t_0, t_1, \dots, t_n) \in \mathbb{R}^{n+1} : \sum_{i=0}^n t_i = 1, \; t_i \ge 0\}\) denote the standard geometric \(n\)-simplex with vertices \(v_0, v_1, \dots, v_n\).

1.2. Define the order-reversing affine homeomorphism \(\iota_n: \Delta^n \to \Delta^n\) by\[\iota_n(t_0, t_1, \dots, t_n) = (t_n, t_{n-1}, \dots, t_0),\]which maps the vertex \(v_i\) to \(v_{n-i}\) for each \(0 \le i \le n\).

1.3. For any continuous map \(\sigma: \Delta^n \to Z\), define the dual simplex map \(\sigma^\vee: \Delta^n \to Z\) by precomposition:\[\sigma^\vee = \sigma \circ \iota_n.\]

2.1. Given a sequence of maps \(X_0 \xrightarrow{f_1} X_1 \xrightarrow{f_2} \cdots \xrightarrow{f_n} X_n\), the iterated mapping cylinder \(M(f_1, \dots, f_n)\) is constructed as the quotient of the disjoint union\[\coprod_{0 \le i_0 < i_1 < \dots < i_k \le n} \Delta^k \times X_{i_0}\]under the face identifications: for \((t_0, \dots, t_k) \in \Delta^k\) with \(t_j = 0\), omit \(t_j\) and \(i_j\); if \(j = 0\), replace \(x \in X_{i_0}\) with \(f_{i_1 i_0}(x) \in X_{i_1}\).

2.2. The dual iterated mapping cylinder \(M^\vee(f_1, \dots, f_n)\) is defined on the same constituent spaces by reflecting each simplex parameter via \(\iota_k\):each cell parameterization \(\sigma: \Delta^k \times X_{i_0} \to M\) is replaced by \(\sigma \circ (\iota_k \times \operatorname{id}_{X_{i_0}})\).

2.3. Under this reflection, the face operators act in reverse order: the \(j\)-th face map \(\partial_j: \Delta^{k-1} \to \Delta^k\) satisfies \(\iota_k \circ \partial_j = \partial_{k-j} \circ \iota_{k-1}\), so the induced structure maps on the dual cylinder satisfy \(d_j^\vee = d_{k-j}\).

3.1. Let \(\Delta X\) be a simplicial space (or complex of spaces) given by a sequence of topological spaces \(X_n = (\Delta X)_n\) for \(n \ge 0\) equipped with face maps \(d_i: X_n \to X_{n-1}\) (\(0 \le i \le n\)) and degeneracy maps \(s_i: X_n \to X_{n+1}\) (\(0 \le i \le n\)) satisfying the simplicial identities.

3.2. Define the dual simplicial space \(\nabla X\) by:\[(\nabla X)_n = (\Delta X)_n \quad \text{for all } n \ge 0,\]with face maps \(d_i^{\nabla}: (\nabla X)_n \to (\nabla X)_{n-1}\) and degeneracy maps \(s_i^{\nabla}: (\nabla X)_n \to (\nabla X)_{n+1}\) defined by\[d_i^{\nabla} = d_{n-i}, \qquad s_i^{\nabla} = s_{n-i} \quad \text{for } 0 \le i \le n.\]

3.3. The dual operators satisfy the simplicial identities: for \(i < j\),\[d_i^{\nabla} d_j^{\nabla} = d_{n-1-i} d_{n-j} = d_{n-j-1} d_{n-i} = d_{j-1}^{\nabla} d_i^{\nabla},\]confirming that \(\nabla X\) is a well-defined simplicial space.