Local homology of degenerate critical points - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:01:01Z http://mathoverflow.net/feeds/question/41128 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41128/local-homology-of-degenerate-critical-points Local homology of degenerate critical points Marco Mazzucchelli 2010-10-05T11:27:42Z 2010-10-05T14:22:12Z <p>Given a smooth function $f:M\rightarrow \mathbb R$ on a manifold, its local homology at a critical point $x$ is the group $$C_\star(x) := H_\star ( M_{ &lt; c} \cup { x } , M_{ &lt; c} ) ,$$ where $H_\star$ denotes singular homology (with any coefficient group), $c=f(x)$, and $M_{ &lt; c}$ is the space of those points $x\in M$ such that $f(x) &lt; c$.</p> <p>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 $j$.</p> <p>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 $j\neq 1$.</p> <p>Does anybody know examples of functions having critical points whose local homology is nonzero in more then one degree?</p> <p>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 $d\neq d_1,...,d_r$?</p> http://mathoverflow.net/questions/41128/local-homology-of-degenerate-critical-points/41156#41156 Answer by Tom Goodwillie for Local homology of degenerate critical points Tom Goodwillie 2010-10-05T14:22:12Z 2010-10-05T14:22:12Z <p>$(x^2+y^2)z^2-c(x^2+y^2+z^2)^2$ for small positive $c$. </p> <p>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 $f=0$.</p>