Let $X,Y,Z$ be manifolds, where $X,Y$ are compact, and $\dim(X)+\dim(Y)=\dim(Z)$. Suppose we have smooth maps $f:X\rightarrow Z$ and $g:Y\rightarrow Z$, we define the mod $2$ intersection number to be $I_2(f,g)=\#(X_{f'}\times_gY)\mod 2$ where $f'$ is any map smoothly homotopic to $f$ and transverse to $g$. We define $\deg_2(f)$ for a map $f:X\rightarrow Y$ where $\dim(X)=\dim(Y)$ to be $I_2(f,i)$ where $i$ is the inclusion map for a single point in $Y$.

My question is; if $X$ and $Y$ are one point spaces, it appears that the degree of the unique map from $X\rightarrow Y$ is equal to $1$, however this seems to contradict the following theorem we were provided in class:

Theorem: if $\deg(f)=1$ then $f$ is not homotopic to the constant map.

I am wondering if I have made a simple error in computing the degree, or if the theorem we were provided needs some extra assumptions on dimension.

  • 1
    $\begingroup$ The theorem needs the assumption that either $X$ or $Y$ have positive dimension. $\endgroup$ – Qiaochu Yuan Nov 23 '17 at 0:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.