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Work by Lizhen Qin shows how to give a CW structure on $M$ such that the interiors of the cells correspond to the unstable manifolds of the gradient flow. The conclusion follows from that. … Qin, On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds, J. Topol. Anal., 2 (2010), no. 4, 469–526. (preprint here) As well as: L. …

Suppose $P, Q$ and $N$ are manifolds. … The displayed map was shown to be $(2n-p-q-3)$-connected (where the lower case letters correspond to the dimensions of the manifolds in question). …

Here is a (slightly) alternative approach in the smooth case. Assume the embedding $Y \to X$ has codimension at least three, where $Y$ is a closed smooth manifold. What we need to know is that the …

In fact there's a whole chapter in the book Burghelea, Dan; Lashof, Richard; Rothenberg, Melvin: Groups of automorphisms of manifolds. With an appendix ("The topological category'') by E. Pedersen. …

Background. Suppose that $M$ is an oriented, connected, closed manifold of dimension $d$ with fundamental class $\mu \in H_d(M;\Bbb Z)$. Let $\Delta : M \to M \times M$ be the diagonal map. Then the p …

I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it …

Here are some definitions: A space is homotopy finite if it is homotopy equivalent to a finite CW complex. A space finitely dominated if it is a retract of a homotopy finite space. A space $X$ is a Po …