Warm-up Problems:

• Compute the homology groups of $S ^ { n }$ for all $n \geq 0$

• Compute the homology of the torus, $T ^ { 2 } = S ^ { 1 } \times S ^ { 1 }$

(1) (Michigan Sept ’10) Consider the 2-dimensional torus T and the topological space

$$
X = T \times [ - 1 , 1 ] / \sim
$$

where $( x , t ) \sim ( x ^ { \prime } , t ^ { \prime } )$ if either $( x , t ) = ( x ^ { \prime } , t ^ { \prime } )$ or $t = t ^ { \prime } \in \{ - 1 , 1 \}$ . Compute $H _ { * } ( X , \mathbb { Z } )$

(2) (Michigan Jan ’10) Let $S ^ { 1 } = \{ z \in \mathbb { C } | | z | | = 1 \}$ . Let X be obtained from the space $S ^ { 1 } \times [ 0 , 1 ]$ by identifying every point $( z , 0 )$ with the point $( z ^ { 4 } , 1 )$ with the quotient topology.
Compute the fundamental group and homology of X.

(3) (Michigan May ’09). Let $S ^ { 1 } = \{ ( x , y , 0 , 0 ) \in \mathbb { R } ^ { 4 } | x ^ { 2 } + y ^ { 2 } = 1 \}$ be the unit circle and consider $M = \mathbb { R } ^ { 4 } \setminus S ^ { 1 }$ . Compute the fundamental group $\pi _ { 1 } ( M )$ and the homology groups H (M) of M.

(4) (Michigan Sept ’08) Compute the homology of the space formed as the union of the unit sphere $\{ ( x , y , z ) | x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 \}$ and the closed interval along the z-axis from $( 0 , 0 , - 1 ) \ \mathrm { { t o } \ ( 0 , 0 , 1 ) }$

(5) (Arizona $\mathrm { A u g } ^ { \prime } 0 6 )$ Compute the singular homology groups $H _ { * } ( X , \mathbb { Z } )$ of the space $X = \mathbb { R } ^ { 3 } \setminus A$ where A is a subset of $\mathbb { R } ^ { 3 }$ homeomorphic to the disjoint union of two unlinked circles.

(6) Use the Meyer-Vietoris sequence to calculate the homology of $X \vee Y$

(7) (Jan ’12) Let X be a space obtained by identifying the three vertices of a standard 2-simplex.

(a) Describe a structure of a ∆-complex on X.

(b) Write down the chain complex corresponding to the ∆-complex in (a)

(c) Compute the homology of X.

(8) (May ’13) Let Y be the standard 3-simplex $\Delta ^ { 3 }$ with a total ordering on its four vertices.
Let X be the ∆-complex obtained from Y by identifying, for each $k \leq 3 .$ , all of its k-dimensional faces such that the identifications respect the vertex ordering.
Thus X has a single k-simplex for each $k \leq 3$ Compute the simplicial homology groups of the ∆-complex X.