HW11

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Homework Assignment 11

Required reading. Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read Hatcher's pages 166-177.

Solve the following problems. (But submit only the underlined ones). In Hatcher's book page 156 problems 9, 12 and 14 and pages 176-177 problems 1, 2, 3, 4, 5, 10 and 11 ( $S^\infty$ and ${\mathbb R}P^\infty$ are defined on Hatcher's pages 6-7).

Due date. This assignment is due at the end of our class on Thursday, April 6, 2006.

Above the line, edit only if you are sure

Feel free to edit below the line

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Problems for Submission

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Hatcher $\S2.2$ , p. 156

9.

Compute the homology groups of the following 2-complexes:

The quotient of $S 2$ obtained by identifying north and south poles to a point.
$S^1\times(S^1 \vee S^1)$ .
The space obtained from $D 2$ by first deleting the interiors of two disjoint subdisks in the interior of $D 2$ and then identifying all three resulting boundary circles together via homeomorphisms preserving clockwise orientations of these circles.
The quotient space of $S^1\times S^1$ obtained by identifying points in the circle $S^1\times\{x_0\}$ that differ by $2π / m$ rotation and identifying points in the circle $\{x_0\}\times S^1$ that differ by $2π / n$ rotation.

12.: Show that the quotient map $S^1\times S^1\to S^2$ collapsing the subspace $S^1\vee S^1$ to a point is not nullhomotopic by showing that it induces an isomorphism on $H 2$ . On the other hand, show via covering spaces that any map $S^2 \to S^1\times S^1$ is nullhomotopic.

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Hatcher $\S2.B$ , p. 176

2.: Show that $\tilde{H}_i(S^n-X) \approx \tilde{H}_{n-i-1}(X)$ when $X$ is homeomorphic to a finite connected graph. (First do the case that the graph is a tree.)
5.: Let $S$ be an embedded $k$ -sphere in $S n$ for which there exists a disk $D^n\subset S^n$ intersecting $S$ in the disk $D^k\subset D^n$ defined by the first $k$ coordinates of $D n$ . Let $D^{n-k}\subset D^n$ be the disk defined by the last $n - k$ coordinates, with boundary sphere $S n - k - 1$ . Show that the inclusion $S^{n-k-1}\hookrightarrow S^n-S$ induces an isomorphism on homology groups.