Find $f$ and $g$ homotopic s.t. induce different homomorphisms Let $X$ and $Y$ be two topological spaces. Are there continuous functions $f,g:X\to Y$ satisfying the following conditions?


*

*$f(a)=g(a)=b$ for some $a\in X$,

*$f$ and $g$ are homotopic, and

*the induced homomorphisms $f_*,g_*:\pi_1(X,a)\to\pi_1(Y,b)$ are different.


As far as I know, if $f$ and $g$ satisfy conditions 1 and 2, then there is a loop $\delta$ at $b$ in $Y$ such that $f_*=F\circ g_*$, where $F:\pi_1(Y,b)\to\pi_1(Y,b)$ is the isomorphism $F([\beta])=[\delta\beta\delta^{-1}]$.
Therefore if $f_*$ and $g_*$ are different then we know that:


*

*$\pi_1(Y,b)$ is not commutative, and

*The homotopy from $f$ to $g$ is not relative to $\{a\}$.


I've been trying with bouquet of circles and some reflections but without getting the desired example.  
 A: You can do this with a wedge of two circles.  Let $\alpha(x), \beta(x): S^1 \to S^1 \vee S^1$ be the two standard generators of $\pi_1(S^1 \vee S^1)$, given by the inclusion of the two summands.  Now, we describe $H: I^2 \to S^1 \vee S^1$ by the following:
$H(s,t) = \left\{ \begin{array}{ll}\beta(t - 4s) & \textrm{if } 0 \le s \le t/4 \\ \alpha\left(\frac{4s-t}{4-2t}\right) & \textrm{if }  t/4 \le s \le 1-t/4 \\ \beta(t + 4s - 4) & \textrm{if } 1 - t/4 \le s \le 1\end{array}
\right.$
This provides a homotopy between elements representing $\alpha$ and $\beta^{-1}\alpha\beta$ in $\pi_1( S^1 \vee S^1)$.  The intuition is to derive the formula from a square with the desired behavior on the edges.  You want the top edge to be $\alpha$ and the bottom edge to be a suitably parametrized version of $\beta^{-1}\alpha\beta$, and the two side edges will be $\beta$.  Form a trapezoid on the interior by connecting the upper corners with the points on the bottom mapped to the basepoint.  At each time $t$, the loop starts at $\beta(t)$, moves to the basepoint, loops around via $\alpha$, and then goes back to $\beta(t)$.
Actually, I think the constructed formula will work for any two noncommuting elements in the fundamental group of any space.
This is part of a standard result on equivalence classes of loops via free homotopies (e.g. see Hatcher Chapter 1 Exercise 6).
