Recent content by HMY

The spirit of my question was is in the sense that M could be infinite dimensional. Could this be proven using Sard's theorem? I also spoke with some other math people a while back and that is what they had suggested. I did look up that there is an infinite dimensional version of Sards...

Let M be a manifold and let f: m -> R a Morse function. Let x be a critical point of f and assume all critical points are non-degenerate. Let W^u(x) be th unstable manifold of x when considering the negative gradient flow on M. Why does the tangent space at x to W^u(x) = Eig^- H^2f(x)...

Let M be a connected manifold. Let E be a submanifold of M of codimension at least 2. Show M\E is connected. I know examples of this result but how can one generally prove it?

What would this be? I'm not fluent with nets (yet) but if this characterization is nice enough then it is perhaps easier for me to work with it here, than what I was trying before.

Let LG be the base point preserving loops (it's a Hilbert manifold). So LG = { f : S^1 -> G s.t. f(0)=1 } where G is a connected, simply connected Lie group. LG is embedded into the (vector space) Hilbert space L^2[0, 2pi] given by f |--> g(t) = f '(t)f(t)^-1 Is LG a closed subset of...

Let H be a separable Hilbert space. What are the continuous linear representations of S^1 on H? I read in an article this is defined as in the finite-dim case. Why is this so? Thanks.

Take the sequence x_n = (1-1/n)e_n in l^2 Consider the map l^2 to R given by x |--> ||x||^2 The set A = {x_n} in l ^2 is closed & its image is not closed in R under the norm topology (it doesn't contain its accumulation point 1). So ultimately the above map is not closed. What I'm...

am I properly making sense of this? Call this map f: (a,b)-->(abx,by/a,bz) f is not injective when you look at (a, b) with b=0 & a not= 0. eg. take another point (c,d) with d=0 & c not= a & c not= 0 So (a,b) not= (c,d). But f(a,b) = (0,0,0) & f(c,d) = (0,0,0)

a 2-torus action on C^3 can be defined by (a,b).(x,y,z)= (abx, a^-1by, bz) What are the orbit type strata of C^3 here? 2-torus can be thought of (S^1)^2. 0 is the only fixed point I can tell, so it's one strata. I just don't understand this seemingly simple action.