All Questions

As described in the title, what is the (topological) Euler characteristic of the homogeneous space $SL_m(\mathbb{C})/SO_m(\mathbb{C})$?

Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. Is it there a formula for ...

One of the corollaries of the Poincaré–Hopf index theorem is that a closed, connected manifold $M$ admits non-vanishing vector fields iff its Euler characteristic is zero; i.e. $\chi(M) = 0$. I am ...

Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with oriented fibers)? For ...

Suppose we have a space M with a real-valued, differentiable function F on M. Under what conditions on F will the Euler characteristic of M be expressed as a (signed) sum of Euler characteristics of ...

Suppose we have a (closed, oriented) 3-manifold M with a Heegard surface F of genus g. Let F* denote F with a puncture. Then the space H of representations of pi_1(F*) on SU(2) is just SU(2)^2g, and ...