# Question on “Homotopy invariance”

i have this from Hatcher's book "Algebric topology"

And i don't understand why $\displaystyle \partial P(\sigma)=\sum_{j\leq i}(-1)^i(-1)^j F\circ (\sigma\times id)|[v_0,...,\widehat{v}_j,...,w_i,...,w_n]+\sum_{j\geq i}(-1)^i(-1)^{j+1} F\circ (\sigma\times id)|[v_0,...,v_i,\widehat{w}_j,...,w_n]$

Plese.

Thank you.

• Have you tried to see why this might be the case for low values of $n$? – Dan Rust Oct 31 '13 at 17:04
• no , but my question is how to calculat $\partial P(\sigma)$ – Vrouvrou Oct 31 '13 at 17:52
• @DanielRust please why ${i-1}$ in $P\partial(\sigma)$ please – Vrouvrou Aug 25 '14 at 15:22

He's just applying the definition and splitting it over two pieces. The first sum is the contribution from the v components (from $\Delta^n$), and the second is running over (is the contribution from) the w components (from I).
• but why there is (-1)^j when $j\leq i$ and $(-1)^{j+1}$when $j\geq i$ ??? – Vrouvrou Nov 1 '13 at 7:52
• @Vrouvrou Carefully check the position that they're in. In particular, look at your definition of $P(\sigma)$ and note how there are two $i$ indices in a row. – zibadawa timmy Nov 4 '13 at 5:39
• please why we have $t-1$ and why strict inequality between $t$ and $j$ in $P\partial(\sigma)=\sum_{t<j} (-1)^t(-1)^j F\circ (\sigma\times id)[v_0,...,v_t,w_t,...,\widehat{w_j},...,w_n]+\sum_{t>j}(-1)^{t-1}(-1)^j F\circ(\sigma\times id)[v_0,...,\widehat{v_j},...,v_t,w_t,...,w_n]$ – Vrouvrou Aug 25 '14 at 15:13