---
title: 18.905 problem set 02
description: attempts and solutions for 18.905 problem set 02
tags:
- course
problems: {}
pageLayout: default
slug: courses/18.905-fall-2016/psets/02
permalink: https://aarnphm.xyz/courses/18.905-fall-2016/psets/02.md
generator:
quartz: v4.6.0
hostedProvider: Cloudflare
baseUrl: aarnphm.xyz
full: https://aarnphm.xyz/llms-full.txt
---
## 5. Acyclic versus contractible chain complexes
> \[!question\] problem 5
>
> (a) Let $A$ be a chain complex (of abelian groups). It is _acyclic_ if $H(A) = 0$, and _contractible_ if it is chain-homotopy-equivalent to the trivial chain complex. Prove that a chain complex is contractible if and only if it is acyclic and for every $n$ the inclusion $Z_n A \hookrightarrow A_n$ is a split monomorphism of abelian groups.
>
> (b) Give an example of an acyclic chain complex that is not contractible.
^problem-5
### attempt
## 6. Product and coproduct in the homotopy category
> \[!question\] problem 6
>
> Propose a construction of the product and the coproduct of two spaces in the homotopy category, and check that your proposal serves the purpose.
^problem-6
### attempt
## 7. Bilinear maps on $\mathbb{Z}S \times \mathbb{Z}T$ and on cyclic groups
> \[!question\] problem 7
>
> (a) Let $S$ and $T$ be sets and $A$ an abelian group. Establish a bijection between the set of maps of sets from $S \times T$ to $A$ and the set of bilinear maps $\mathbb{Z}S \times \mathbb{Z}T \to A$.
>
> (b) For positive integers $m, n$, let $\mathbb{Z}/m$, $\mathbb{Z}/n$ denote the cyclic groups of order $m, n$. Construct a surjective bilinear map $\mu : \mathbb{Z}/m \times \mathbb{Z}/n \to \mathbb{Z}/\gcd\{m, n\}$. Show that any bilinear map $\mathbb{Z}/m \times \mathbb{Z}/n \to A$ factors uniquely as $f \circ \mu$ where $f : \mathbb{Z}/\gcd\{m, n\} \to A$ is a homomorphism.
^problem-7
### attempt
## 8. Splittings of short exact sequences; ladders of long exact sequences
> \[!question\] problem 8
>
> (a) Let $0 \longrightarrow A \xrightarrow{\;i\;} B \xrightarrow{\;p\;} C \longrightarrow 0$ be a short exact sequence. Show that the following three sets are in bijection with one another.
>
> (i) The set of homomorphisms $\sigma : C \to B$ such that $p\sigma = 1_C$.
>
> (ii) The set of homomorphisms $\pi : B \to A$ such that $\pi i = 1_A$.
>
> (iii) The set of homomorphisms $\alpha : A \oplus C \to B$ such that $\alpha(a, 0) = ia$ for all $a \in A$ and $p\alpha(a, c) = c$ for all $(a, c) \in A \oplus C$.
>
> Moreover, show that any homomorphism as in (iii) is an isomorphism.
>
> Any one of these structures is a _splitting_ of the short exact sequence, and the sequence is then said to be _split_.
>
> (b) Suppose that
>
> source code
>
> is a “ladder”: a map of long exact sequences. So both rows are exact and each square commutes. Suppose also that every third vertical map is an isomorphism, as indicated. Prove that these data naturally determine a long exact sequence
>
> $$
> \cdots \longrightarrow A_n \longrightarrow A'_n \oplus B_n \longrightarrow B'_n \longrightarrow A_{n-1} \longrightarrow \cdots
> $$
^problem-8
### attempt
## 9. $3 \times 3$ lemma; long exact sequence of a triple
> \[!question\] problem 9
>
> (a) (”$3 \times 3$ lemma.”) Let
>
> source code
>
> be a commutative diagram of abelian groups. Assume that all three columns are exact, that all but one of the rows is exact, and that the compositions in the remaining row are trivial. Prove that the remaining row is also exact. (Hint: view each row as a chain complex …)
>
> (b) (“Long exact homology sequence of a triple.”) Let $(C, B, A)$ be a “triple,” so $C$ is a space, $B$ is a subspace of $C$, and $A$ is a subspace of $B$. Show that there are natural transformations $\partial : H_n(C, B) \to H_{n-1}(B, A)$ such that
>
> $$
> \cdots \longrightarrow H_n(B, A) \xrightarrow{\;i_*\;} H_n(C, A) \xrightarrow{\;j_*\;} H_n(C, B) \xrightarrow{\;\partial\;} H_{n-1}(B, A) \longrightarrow \cdots
> $$
>
> is exact, where $i : (B, A) \to (C, A)$ and $j : (C, A) \to (C, B)$ are the inclusions of pairs. (**9 (a)** might be useful.)
^problem-9
### attempt
## 10. Cone and suspension; $\widetilde{H}_{n-1}(X) \cong H_n(SX, *)$
> \[!question\] problem 10
>
> This exercise generalizes our computation of the homology of spheres, and introduces several important constructions.
>
> The _cone_ on a space $X$ is the quotient space $CX = X \times I / X \times \{0\}$, where $I$ is the unit interval $[0, 1]$. The cone is a pointed space, with basepoint $*$ given by the “cone point,” i.e. the image of $X \times \{0\}$. (By convention, the cone on the empty space $\varnothing$ is a single point, the cone point.) Regard $X$ as the subspace of $CX$ of all points of the form $(x, 1)$.
>
> Define the _suspension_ of a space $X$ to be $SX = CX/X$. Make $SX$ a pointed space by declaring the image of $X \subseteq CX$ to be the basepoint in $SX$. (By convention, the quotient $W/\varnothing$ is the disjoint union of $W$ with a single point, which is declared to be the basepoint. So $S\varnothing = */\varnothing$ is the discrete two-point space, with the new point as basepoint.)
>
> The quotient map induces a map of pairs $f : (CX, X) \to (SX, *)$.
>
> (a) Show that $CX$ is contractible.
>
> (b) Show that there is a natural isomorphism $\widetilde{H}_{n-1}(X) \to H_n(SX, *)$, for any $n$.
^problem-10
### attempt
