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Thom's first isotopy lemma says that given a smooth map $f:M\to P$ between smooth manifolds, and a closed Whitney stratified subset $S$ of $M$, such that $f|_S:S\to P$ is proper and $f|_X:X\to P$ is a …

I have a very concrete question. Let $N={\Bbb C}-\{\pm 1\}$, and ${\Bbb Z}_2$ is acting on $N$ by rotation around $0$. Consider $M=\{(x,y)\in N^2\ |\ {\Bbb Z}_2x\neq {\Bbb Z}_2y\}$. Let $p:M\to N$ be …

Let $M$ and $N$ be two connected, smooth (possibly non-compact), aspherical manifolds (that is, $\pi_k(-)=1$ for $k\geq 2$) of dimensions $n$ and $n-r$ respectively. Let $f:M\to N$ be a smooth map ind …

Let $p:E\to B$ be a locally trivial fibration of connected, non-compact smooth manifolds. Let $U\subset E$ be a connected open subset and $p|_U:U\to p(U)$ has connected diffeomorphic fibers. Can we co …

Let $p:E\to B$ be a locally trivial fibration of connected, non-compact smooth manifolds. Let $U\subset E$ be a connected open subset and $p|_U:U\to p(U)$ has diffeomorphic fibers. Can we conclude tha …

For motivation I recall two classical examples of locally trivial fiber bundle projections, due to Brieskorn (in `Sur les groupes de tresses'). ${\Bbb C}^n-\cup_{i\neq j}\{(z_1,z_2,\ldots ,z_n): z_i= …

Let $M$ be a Seifert fibered space over an orbifold $B$. Assume that $B$ is good and has infinite orbifold fundamental group. Then it is well known that there is a short exact sequence. $$1\to {\Bbb Z …

How does one check that the following space is aspherical? $X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$. One way I can think of is to give …