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\begin{document}
\parindent 0in

\parbox[b]{2in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2526}{2025-26}: \href{http://drorbn.net/26-1301}{MAT 1301 Algebraic Topology}:
}~{\LARGE\bf Homework Assignment 4}
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  \null\hfill\sheeturl
  \newline\null\hfill Due February 25 at 11:59pm
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\noindent{\bf Problem 1.} For any based space $(X,x_0)$ define a natural non-zero map $\alpha_{(X,x_0)}\colon\pi_1(X,x_0)\to H_1(X)$. The challenge will be to show that your definition is well-defined.

What means ``non-zero''? We don't have the tools yet to prove that $H_1$ is ever non-zero! So I will be happy enough with a map that is non-zero as per our intuitive notion of $H_1(S^1)$.

What means ``natural''? That if $f\colon(X,x_0)\to(Y,y_0)$, then the following diagram is commutative:
\[ \xymatrix@C=2cm{
  \pi_1(X,x_0) \ar[r]^{\alpha_{(X,x_0)}} \ar[d]_{f_*} & H_1(X) \ar[d]_{f_*} \\
  \pi_1(Y,y_0) \ar[r]^{\alpha_{(Y,y_0)}} & H_1(Y)
} \]
(In other words, $\alpha$ should be a ``natural transformation'').

\vskip 2mm
\noindent{\bf Problem 2.} Show that the set $\Delta_n'\coloneqq\{(s_1,s_2,\ldots,s_n)\colon 0\leq s_1\leq s_2\leq\dots\leq s_n\leq 1\}\subset I^n\subset\bbR^n$ is homeomorphic to the standard $n$ simplex $\Delta_n$ via a map of the form $[v_0,\ldots,v_n]\colon\Delta_n\to\Delta'_n$ where $v_0,\ldots,v_n\in I^n$.

\vskip 2mm
\noindent{\bf Problem 3.} Using the alternative model of the $n$-simplex presented in the previous problem, show that
\begin{enumerate}
\item The $n$-cube $I^n$ can be presented as a union of size $n!$ of $n$-simplices.
\item The product $\Delta_p\times\Delta_q$ can be presented as a union of size $\binom{p+q}{q}$ of $(p+q)$-simplices.
\end{enumerate}

\vskip 2mm
\noindent{\bf Problem 4.} ``Homotopies between maps'' define an ``ideal'' within the category of topological spaces and continuous maps between them: the homotopy relation is an equivalence relation, and if $f_1\sim f_2$, then $f_1\circ g\sim f_2\circ g$ and $g\circ f_1\sim g\circ f_2$ whenever these compositions make sense. Show that the same is true for the notion ``homotopy of morphisms between chain complexes'', within the category Kom of chain complexes.

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