Given a smooth function $f:M\rightarrow \mathbb R$ on a manifold, its local homology at a critical point $x$ is the group
$$ C\_\star(x) := H\_\star ( M\_{ < c} \cup \{ x \} , M\_{ < c} ) ,$$ 
where $H\_\star$ denotes singular homology (with any coefficient group), $c=f(x)$, and $M\_{ < c}$ is the space of those points $x\in M$ such that $f(x) < c$.


If $x$ is a non-degenerate critical point, then $C\_\star(x)$ is completely determined by the Morse index of $f$ at $x$: the group $C\_j(x)$ is equal to the coefficient group of the homology for $j=\mathrm{ind} (x)$, and is trivial for other values of $j$.


If $x$ is degenerate, the knowledge of $\mathrm{ind}(x)$ and $\mathrm{nul}(x)$ (this latter being the nullity of $f$ at $x$) is not enough to determine $C\_\star (x)$. It is easy to build examples of functions on $\mathbb R^2$ having a critical point $x$ with local homology $C\_1(x)= G\oplus ...\oplus G$ ($k$ times, where $k>1$ and $G$ is the coefficient group) and $C\_j(x)=0$ for $j\neq 1$. For instance, consider the function $f:\mathbb R^2\rightarrow\mathbb R$ given by
$$ f(x,y)=(y-2x^2)(y-x^2)(y+x^2)(y+2x^2). $$
Here, the origin is a critical point whose local homology (say, with $\mathbb Z\_2$ coefficients) should be $C\_1(0)=\mathbb Z\_2\oplus\mathbb Z\_2\oplus \mathbb Z\_2$ and $C\_j(0)=0$ for $j\neq 1$.


Does anybody know examples of functions having critical points whose local homology is nonzero in more then one degree?


If the answer to the previous question is yes (as I would expect), is it true that given $(n\_1,d\_1), ..., (n\_r,d\_r)$ there exists a function $f:M\rightarrow\mathbb R$ with a critical point $x$ whose local homology is given by $C\_{d\_j}(x)=G^{ \oplus n\_j }$ and $C\_d(x)=0$ for $d\neq d\_1,...,d\_r$?



Local homology of degenerate critical points
$(x^2+y^2)z^2-c(x^2+y^2+z^2)^2$ for small positive $c$. 


More generally $f-cr^{2d}$ where $f\ge 0$ is a homogeneous polynomial function of degree $2d$ in $n$ variables. The local homology at the origin should be essentially the homology of the set of points in $S^{n-1}$ where $f=0$.


