Simplicial and Delta Complexes

Suppose that a complex \(X\) is the union of two subcomplexes \(A\) and \(B\) with \(A\), \(B\), and \(A \cap B\) contractible. Prove that \(X\) is contractible.

Let \(\Delta ^n\) be the \(n\)-simplex with its usual \(\Delta \)-complex structure. Find all subcomplexes \(X\) of \(\Delta ^n\) such that \(\tilde {H}_{n-1}(X)\) is non-zero.

Suppose \(X\) and \(Y\) are cell complexes and that \(A\subseteq X\) is a subcomplex. Let \(f:A\to Y\) be continuous and let \(Y\sqcup _f X\) be \(Y\) with \(X\) attached along \(A\) via \(f\). Prove that \(Y\sqcup _f X\) is homotopy equivalent to a cell complex.

Let \(\{B_n\}_{n\in \mathbb {Z}}\) and \(\{C_n\}_{n\in \mathbb {Z}}\) be chain complexes. What is the definition of a chain map from \(\{B_n\}_{n\in \mathbb {Z}}\) to \(\{C_n\}_{n\in \mathbb {Z}}\)?

Regard the \(n\)-simplex \(\Delta ^n\) as a \(\Delta \)-complex in the natural way, with \((n+1)\) vertices and a single \(n\)-simplex. Find all subcomplexes \(X\subset \Delta ^n\) such that \(\operatorname {H}_{n-1}(X)\ne 0\) (with a proof that there are no others).

Let \(p:\widetilde {X}\to X\) be an \(n\)-sheeted covering map between two connected, locally path-connected spaces. Let \(C_*(X)\) and \(C_*(\widetilde {X})\) denote the singular chain complexes of \(X\) and \(\widetilde {X}\) respectively.

1. Give a definition of a chain morphism and prove that \(\tau :C_*(X)\to C_*(\widetilde {X})\) given by \[ \tau (\sigma )=\sum _{i=1}^n \widehat {\sigma }_i \] is a chain morphism, where \(\widehat {\sigma }_1,\ldots ,\widehat {\sigma }_n\) are the \(n\) lifts of \(\sigma \) to \(\widetilde {X}\), and \(\sigma \) is a singular simplex in \(X\).
2. Prove that \[ p_*:H_i(\widetilde {X};\mathbb {Q})\to H_i(X;\mathbb {Q}) \] is always surjective. Hint: Consider the composition \(p_*\tau :C_*(X)\to C_*(X)\).
3. Find an example where \[ p_*:H_i(\widetilde {X};\mathbb {Z})\to H_i(X;\mathbb {Z}) \] is not surjective.

Let \(K\) be the \(\Delta \)-complex obtained by identifying all the 2-dimensional faces of a 3-simplex. Calculate the simplicial homology of \(K\).

Construct a \(\Delta \)-complex for the Klein bottle and calculate its simplicial homology for both \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

True or false: Suppose that \(X\) is a finite CW-complex and \(A\subset X\) is a subcomplex. Then the relative homotopy groups \(\pi _j(X,A)\) are isomorphic to the homotopy groups \(\pi _j(X/A)\), where \(X/A\) is the CW-complex obtained from \(X\) by collapsing \(A\) to a point. Justify your answer.

Let \(K\) denote a finite simplicial complex. The Euler characteristic is given by \[ \chi (K)=\sum _{j\ge 0}(-1)^j\alpha _j \] where \(\alpha _j\) is the number of \(j\)-simplices in \(K\). If \(\mathrm {Sd}\,K\) denotes barycentric subdivision then \(\chi (K)=\chi (\mathrm {Sd}\,K)\).