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The fundamental group

Loops up to deformation; the circle's winding number; van Kampen.

Loops based at a point, composed by concatenation and identified up to deformation, form a group. For the circle it is ℤ — the winding number — and for the plane minus a point likewise. Picture it: a loop winding twice around the hole cannot be shrunk without crossing it. Think it: π₁ is a functor: continuous maps induce homomorphisms, which is how topology becomes algebra.

Contoh yang berhasil: polar form of -1

Precalculus · Complex-Numbers

Polar form of -1

-1

Langkah demi langkah

  1. z = -1 = -1 + (0)i

    Read off the real and imaginary parts.

  2. |z| = \sqrt{-1^2 + 0^2} = 1

    The modulus is the distance from the origin (Pythagoras).

  3. \theta = \arg z = \pi \approx 3.1416

    The argument is the angle from the positive real axis (watch the quadrant).

  4. z = 1\left(\cos \pi + i\sin \pi\right) = 1 e^{i \pi}

    Polar and exponential forms (Euler).

Mengungkapkan jawabannya
1 e^{i \pi}

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\theta
theta
The usual name for an angle.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
i
imaginary unit
i² = −1.
\approx
approximately equal
Equal to the precision shown, not exactly.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
G \cong H,\ G / N
isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.
\pi_1(X),\ H_n(X),\ \chi
fundamental group, homology, Euler characteristic
Loops up to deformation; holes in each dimension; V − E + F.

How to: The fundamental group

  1. Read off the real and imaginary parts.
  2. The modulus is the distance from the origin (Pythagoras).
  3. The argument is the angle from the positive real axis (watch the quadrant).
  4. Polar and exponential forms (Euler).

Questions people ask

What does the fundamental group measure?

Loops in a space up to deformation. In a plane every loop shrinks to a point (trivial group); around a hole a loop can wind n times (the integers).

Cobalah sendiri

Lebih dalam Algebraic Topology

Homology and the Euler characteristic