Exercises

1. Fill in the details in the computation of the fundamental group of the projective plane, Klein's bottle and the torus done in the lecture by providing a careful proof of equations (26.8), (26.10) and (26.11). Hint: Use polar coordinates. Continuously shrink the path $\beta$ to the point $x_0$.
2. Show that the fundamental group of the wedge of $n$ copies of $S^1$ is the free group on $n$ generators. Calculate the fundamental group of the truncated grid
   $$\{(x,y)\in \mathbb{R}^2/ x\in \mathbb{Z}$$ or $$y \in \mathbb{Z},\; 0\leq x \leq n,\; 0 \leq y \leq n\}.$$

3. Determine the generators of double torus by expressing it as a union of open sets each of which is a torus from which a tiny closed disc has been removed.
4. Let $C$ be the union of the two unlinked circles
   

   $$(x-2)^2 + y^2 = 1,\; z = 0,$$

   $$(x+2)^2 + y^2 = 1,\; z = 0.$$

   
   in $\mathbb{R}^3$. Show that $\pi_1(\mathbb{R}^3 - C)$ is the free group on two generators.
5. Calculate the fundamental groups of the following spaces

   (i)

   $\mathbb{R}^4$ minus a line.

   (ii)

   $\mathbb{R}^4$ minus a two dimensional linear subspace.

   (iii)

   $\mathbb{R}^4$ minus two parallel lines.

   (iv)

   $\mathbb{R}^4$ minus two intersecting lines.

   (v)

   $\mathbb{R}^3$ minus the coordinate axes

   (vi)

   $\mathbb{C}^2 - \{(z_1, z_2)/ z_1z_2 = 0\}$

   (vii)

   $\mathbb{R}^3$ minus finitely many points.

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Lecture - XXVII (Test IV)

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1. Use the Seifert Van Kampen theorem to compute the fundamental group of the double torus.
2. Let $K$ be a compact subset of $\mathbb{R}^3$ and regard $S^3$ as the one point compactification of $\mathbb{R}^3$. Show that $\pi_1(\mathbb{R}^3 - K) = \pi_1(S^3 - K)$.
3. If $C$ is the circle in $\mathbb{R}^3$ given by the pair of equations
   $$x^2 + z^2 = 1,\quad z= 0,$$
   show that $\pi_1(\mathbb{R}^3 - C) = \mathbb{Z} \oplus \mathbb{Z}$. Let $C^{\prime}$ be the circle given by
   $$(y-1)^2 + z^2 + 1,\quad x = 0.$$
   Show that $\pi_1(\mathbb{R}^3 - C\cup C^{\prime}) = \mathbb{Z}\oplus \mathbb{Z}$. Hint: Use stereographic projection.
4. Show that the complement of a line in $\mathbb{R}^4$ is simply connected.
5. Calculate the fundamental group of $\mathbb{C}^2 - \{(z_1, z_2)/ z_1z_2 = 0\}$.

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Lecture - XXVIII Introductory remarks on homology theory

in In the first part of the course we focused on the fundamental group and its basic properties. We discussed an elegant solution of the lifting problem for covering projections in terms of the fundamental group. While the theory of fundamental groups and covering spaces is fairly adequate for many applications in low dimensional geometry and other parts of mathematics such as the theory of function of one complex variable, it is quite ineffective when higher dimensional objects are involved. For instance the ball $B^n$ and the sphere $S^{n-1}$ both have trivial fundamental group ($n \geq 3$) which renders it useless for proving the higher dimensional analogues of the Brouwer's fixed point theorem.

Homology theory provides a functor that is quite convenient for understanding the geometry of ``higher dimensional objects'' which has the added advantage of being easily computable (at least for a large class of interesting spaces). While the fundamental group functor respect products, the homology groups of $X\times Y$ are not so easily described in terms of the homology groups of $X$ and $Y$. A covering projection is a very special case of a fiber bundle with discrete fibers. We have seen that in the case of a covering projection $p: {\tilde X}\longrightarrow X$ we have a relationship between $\pi_1(X)$ and $\pi_1({\tilde X}).$ The story is decidedly more complicated with homology groups. For instance some work is required to compute the homology groups of the real projective spaces $\mathbb{R}P^n$. Homotopy theory is better suited for studying fibrations where the use of homology would entail the formidable machinery of ``spectral sequences''. However, on the computational side there is a very useful substitute for the Seifert Van Kampen theorem in homology known as the Mayer Vietoris sequence. We shall use it to calculate efficiently the homology groups of a large number of spaces.

There are several approaches to the homology theory, the oldest being the simplicial theory. Homology theory evolved over several decades through the early part of the twentieth century becoming progressively abstract.

The theory we discuss in this course is known as the singular homology theory and would appear somewhat non-intuitive in the beginning but we hope that the examples and applications presented would enable the students to digest the material. Singular homology theory appeared rather late in the development of algebraic topology and is a culmination of efforts spanning a few decades by several eminent topologists. In the intervening years several seemingly different homology theories developed the oldest and most intuitive being simplicial homology theory that applies to the restricted class of simplicial complexes. However the topological invariance is highly non-trivial and beset with technical complications.